analysis, modeling, and simulation of the tides in the

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Analysis, Modeling, And Simulation Of The Tides In The Analysis, Modeling, And Simulation Of The Tides In The

Loxahatchee River Estuary (Southeastern Florida). Loxahatchee River Estuary (Southeastern Florida).

Peter Bacopoulos University of Central Florida

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ANALYSIS, MODELING, AND SIMULATION OF THE TIDES IN THE LOXAHATCHEE RIVER ESTUARY (SOUTHEASTERN FLORIDA)

by

PETER BACOPOULOS B.S. University of Central Florida, 2003

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science

in the Department of Civil and Environmental Engineering in the College of Engineering and Computer Science

at the University of Central Florida Orlando, Florida

Fall Term 2005

ABSTRACT

Recent cooperative efforts between the University of Central Florida, the Florida Department of

Environmental Protection, and the South Florida Water Management District explore the

development of a two-dimensional, depth-integrated tidal model for the Loxahatchee River

estuary (Southeastern Florida). Employing a large-domain approach (i.e., the Western North

Atlantic Tidal model domain), two-dimensional tidal flows within the Loxahatchee River estuary

are reproduced to provide: 1) recommendations for the domain extent of an integrated,

surface/groundwater, three-dimensional model; 2) nearshore, harmonically decomposed, tidal

elevation boundary conditions.

Tidal simulations are performed using a two-dimensional, depth-integrated, finite

element-based code for coastal and ocean circulation, ADCIRC-2DDI. Multiple variations of an

unstructured, finite element mesh are applied to encompass the Loxahatchee River estuary and

different spatial extents of the Atlantic Intracoastal Waterway (AIW). Phase and amplitude

errors between model output and historical data are quantified at five locations within the

Loxahatchee River estuary to emphasize the importance of including the AIW in the

computational domain. In addition, velocity residuals are computed globally to reveal

significantly different net circulation patterns within the Loxahatchee River estuary, as

depending on the spatial coverage of the AIW.

ii

ACKNOWLEDGEMENT

The study presented herein is the product of my individual efforts combined with the assistance

received from a number of people, to whom I would like to express appreciation and gratitude

for helping me complete this work. First, I would like to thank my advisor, Dr. Scott C. Hagen,

for providing me with the opportunity to pursue graduate study at the University of Central

Florida. This thesis would not have been possible without his support and guidance, in addition

to the knowledge and education that I received firsthand by studying under his direction. I

would also like to thank Drs. Manoj Chopra and Gour-Tsyh Yeh for agreeing to serve on my

thesis committee and for offering a multitude of interesting and challenging courses; my

Japanese lab mates, Yuji Funakoshi and Satoshi Kojima, for sharing their pleasant culture with

all of us in the lab; present lab members, David Coggin and Mike Salisbury, for their

perspectives and advice; former lab members, Daniel Dietsche, Ryan Murray, and Michael

Parrish, for their contributions; Dr. Gordon Hu and other South Florida Water Management

District (SFWMD) members for coordinating a boat trip (site visit) along the Loxahatchee River;

Dr. Alan Zundel and his students in the Environmental Modeling Research Laboratory at

Brigham Young University for providing me with specialized training in mesh generation; and

the University of Central Florida campus and community for making my graduate education

quite an enjoyable experience.

In the grander scheme of things, I am very fortunate to have been blessed with a loving

and supporting family. I send the deepest appreciation to my parents for their many sacrifices

and for instilling the principles and values that have allowed me to succeed in life. To my twin

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brother, many thanks for being there during the good times and for helping me through the rough

times. It is through our continued efforts to help each other that we grow stronger individually

and together as a family unit.

This study is funded in part by the SFWMD under Contract No. CC11704A and the

Florida Department of Environmental Protection (FDEP) under Contract No. S0133. The

statements, findings, conclusions, and recommendations expressed herein are those of the author

and do not necessarily reflect the views of the SFWMD, FDEP, or its affiliates.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi CONVERSION FACTORS AND PHYSICAL CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . xxiii DATUM TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2. TIDAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER 3. LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1. RECENT PROGRESS IN THE TWO- AND THREE-DIMENSIONAL

MODELING OF TIDES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2. PREVIOUS MODELING STUDIES FOR THE LOXAHATCHEE RIVER ESTUARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3. TIDAL ASYMMETRY AND RESIDUAL CIRCULATION . . . . . . . . . . . . . 42

CHAPTER 4. NUMERICAL MODEL DOCUMENTATION . . . . . . . . . . . . . . . . . . . . . . . . 48

CHAPTER 5. PRESENTATION OF STUDY AREA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

CHAPTER 6. PRELIMINARY MODELING EFFORTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.1. WNAT MODEL DOMAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2. FINITE ELEMENT MESH DEVELOPMENT (PRELIMINARY VERSION) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

v

6.3. MODEL INITIALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4. PRELIMINARY MODEL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.5. MODEL-SENSITIVITY RUNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

CHAPTER 7. DOMAIN SPECIFICATION AND FINAL COMPUTATIONAL MESH . . 107

7.1. FINITE ELEMENT MESH DEVELOPMENT (SECOND GENERATION) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2. IMPROVED MODEL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.3. FINAL COMPUTATIONAL MESH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

CHAPTER 8. CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

APPENDIX A. TIDAL POTENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

APPENDIX B. NODAL CYCLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

APPENDIX C. HISTORICAL WATER SURFACE ELEVATIONS AND RESYNTHESIZED HISTORICAL TIDAL SIGNALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

APPENDIX D. TIDAL CONSTITUENT AMPLITUDE AND PHASE LISTING . . . . . . . . 178

APPENDIX E. COMPUTED METEOROLOGICAL RESIDUALS AND RESYNTHESIZED SEASONAL VARIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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LIST OF TABLES

Table 2.1. The basic speeds and origins of the astronomical arguments that give the frequencies of the harmonic components (after Harris [1991]) . . . . . . . . . . . . . . . . 9

Table 2.2. The dominant harmonics of the tides and their physical causes (after Reid [1990]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Table 2.3. 68 tidal constituents and corresponding nodal adjustment factors extracted by T_TIDE and used in the resynthesis of the historical tidal signal . . . . . . . . . . . . . 23

Table 2.4. Computed form factors associated with the tides in the Loxahatchee River estuary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Table 3.1. Tidal asymmetry in the Loxahatchee River estuary, represented in terms of the M2- M4 tidal constituent interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Table 3.2. Magnitude of the tidal asymmetry in the Loxahatchee River estuary, represented in terms of the ha dimensionless parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 5.1. 7 major drainage sub-basins of the Loxahatchee River watershed (after FDEP [1998]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Table 6.1. Characteristics of the WNAT model domain-based finite element meshes . . . . . . 83

Table 6.2. 23 tidal constituents employed by ADCIRC-2DDI . . . . . . . . . . . . . . . . . . . . . . . . 90

Table 6.3. Errors associated with the preliminary model results, in correspondence to the tidal resynthesis plots presented in Figure 6.5-6.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Table 6.4. Absolute average phase errors (°) associated with the first set of model-sensitivity runs. The lowest absolute average phase errors are bolded in order to highlight the

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best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Table 6.5. Coefficients of determination (-) (see Eq. [6.1]) associated with the first set of model-sensitivity runs. The highest values of the coefficient of determination are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . 101

Table 6.6. Normalized RMS errors (-) (see Eq. [6.2]) associated with the first set of model- sensitivity runs. The lowest normalized RMS errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Table 6.7. Absolute average phase errors (°) associated with the second set of model- sensitivity runs. The lowest absolute average phase errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Table 6.8. Coefficients of determination (-) (see Eq. [6.1]) associated with the second set of model sensitivity runs. The highest values of the coefficient of determination are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . 105

Table 6.9. Normalized RMS errors (-) (see Eq. [6.2]) associated with the second set of model- sensitivity runs. The lowest normalized RMS errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table 7.1. Hydrodynamic measurements associated with the additional inlets described by the second generation of the finite element mesh (after Carr de Betts [1999]) . . . . . 111

Table 7.2. Absolute average phase errors (°) associated with the application of the second generation of the finite element mesh. The lowest absolute average phase errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . 114

Table 7.3. Coefficients of determination (-) (see Eq. [6.1]) associated with the application of the second generation of the finite element mesh. The highest values of the

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coefficient of determination are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Table 7.4. Normalized RMS errors (-) (see Eq. [6.2]) associated with the application of the second generation of the finite element mesh. The lowest normalized RMS errors are bolded in order to highlight the best performing model results . . . . . . . . . . . 115

Table 7.5. Absolute average phase errors (°) associated with the preliminary model runs and application of the second generation of the finite element mesh (both for

). The lowest absolute average phase errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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Table 7.6. Coefficients of determination (-) (see Eq. [6.1]) associated with the preliminary model runs and application of the second generation of the finite element mesh (both for ). The highest values of the coefficient of determination are bolded in order to highlight the best performing model results . . . . . . . . . . . 116

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Table 7.7. Normalized RMS errors (-) (see Eq. [6.2]) associated with the preliminary model runs and application of the second generation of the finite element mesh (both for

). The lowest normalized RMS errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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Table 7.8. Absolute average phase errors (°) associated with the application of the final version of the finite element mesh. The lowest absolute average phase errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . 122

Table 7.9. Coefficients of determination (-) (see Eq. [6.1]) associated with the application of the final version of the finite element mesh. The highest values of the coefficient of

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determination are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Table 7.10. Normalized RMS errors (-) (see Eq. [6.2]) associated with the application of the final version of the finite element mesh. The lowest normalized RMS errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . 123

Table 7.11. Absolute average phase errors (°) associated with the applications of the second generation and final version of the finite element mesh (both for ). The lowest absolute average phase errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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Table 7.12. Coefficients of determination (-) (see Eq. [6.1]) associated with the applications of the second generation and final version of the finite element mesh (both for

). The highest values of the coefficient of determination are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . 124

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Table 7.13. Normalized RMS errors (-) (see Eq. [6.2]) associated with the applications of the second generation and final version of the finite element mesh (both for

). The lowest normalized RMS errors are bolded in order to highlight the best performing model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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Table D.1. 68 tidal constituent amplitudes and phases extracted by T_TIDE and used in the resynthesis of the historical tidal signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

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LIST OF FIGURES

Figure 1.1. Map of the Loxahatchee River estuary, including the locations of the five water level gaging stations (Coast Guard Dock, Pompano Drive, Boy Scout Dock, Kitching Creek, and River Mile 9.1, corresponding to the circles numbered 1-5, respectively) situated within its interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Figure 2.1. (a) Spring tide conditions when the Moon is in syzygy and (b) neap tide conditions when the Moon is in quadrature (after Pugh [2004]) . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 2.2. Frequency-dependent pattern of the (a) diurnal and (b) semi-diurnal tidal constituents with their associated equilibrium amplitudes plotted on a logarithmic scale (after Cartwright and Edden [1973]). Each individual vertical line represents a tidal constituent; note the clustering of tidal constituents into groups within each tidal species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 2.3. Computed form factors associated with the tides in the WNAT model domain, highlighting the diurnal (FF 3.00) and semi-diurnal (FF = 0.00 – 0.25) tidal regimes experienced within the Gulf of Mexico and Caribbean Sea and in the western North Atlantic Ocean, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

≥

Figure 3.1. Distortion of a tidal wave propagating through shallow water up a channel in the positive x-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 4.1. Depth-dependence of the hybrid bottom friction factor; see Eq. (4.21) . . . . . . . . . 59

Figure 5.1. Map of the Loxahatchee River watershed (after FDEP [1998]) highlighting (a) the boundaries of JDSP and the Loxahatchee and Hungryland Sloughs and the layout of

xi

the local road/highway system along with (b) the margins of the seven major drainage sub-basins located within its interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 5.2. (a) Bathymetry (displayed in meters below MSL) of the Loxahatchee River estuary with river-kilometer distances plotted along the Loxahatchee River including (b) its associated river bottom profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Figure 6.1. Bathymetry (displayed in meters below MSL) of the WNAT model domain, highlighting the open-ocean boundary and the areas of the continental shelf break (183 m) and the edge of Blake’s Escarpment (1200 m) (boxes 1 and 2, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 6.2. LTEA-based finite element mesh of Kojima (2005), highlighting the increased grid resolution remaining over the areas of the continental shelf break and the edge of Blake’s Escarpment (red box) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 6.3. Coastline and bathymetric definition of the Loxahatchee River estuary, as represented by the current version of the integrated, three-dimensional estuary model (after Yeh et al. [2004]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 6.4. Spatial discretization of the Loxahatchee River estuary: (a) finite element mesh representation and (b) its associated nodal density (displayed in meters) . . . . . . . 87

Figure 6.5. Resyntheses of (preliminary) model (red solid line) and historical (blue solid line) tidal constituents, corresponding to the water level gaging station located at Coast Guard Dock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 6.6. Resyntheses of (preliminary) model (red solid line) and historical (blue solid line) tidal constituents, corresponding to the water level gaging station located at

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Pompano Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 6.7. Resyntheses of (preliminary) model (red solid line) and historical (blue solid line) tidal constituents, corresponding to the water level gaging station located at Boy Scout Dock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Figure 6.8. Resyntheses of (preliminary) model (red solid line) and historical (blue solid line) tidal constituents, corresponding to the water level gaging station located at Kitching Creek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 6.9. Resyntheses of (preliminary) model (red solid line) and historical (blue solid line) tidal constituents, corresponding to the water level gaging station located at River Mile 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 7.1. (a) Extension (black solid line) of the preliminary boundary (red solid line), including the domain extent of the final version of the finite element mesh (dashed inset box). The blue inset boxes relate to Figure 7.2. (b) Spatial discretization associated with the second generation of the finite element mesh. The green inset boxes relate to Figure 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 7.2. The entrance to the Indian River Lagoon and the relatively narrow channels of the AIW continuing (a) north and (b) south, respectively, of the extended boundary (red solid line) (see blue inset boxes of Figure 7.1[a]). USGS aerial photography is supplied by TerraServer-USA (http://teraserver.microsoft.com/; website accessed on December 16, 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 7.3. (a,d,g) Boundary definition, (b,e,h) spatial discretization, and (c,f,i) bathymetry (displayed in meters below MSL) associated with the second generation of the finite element mesh, for the regions surrounding Fort Pierce, St. Lucie, and Lake Worth

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Inlets, respectively (see green inset boxes of Figure 7.1[b]). USGS aerial photography is supplied by TerraServer-USA (http://teraserver.microsoft.com/; website accessed on December 16, 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 7.4. (a,c) Vectors and (b,d) magnitudes (cm/s) of the residual circulation occurring through Jupiter Inlet and the north arm of the AIW, as based on the application of the second generation of the finite element mesh and preliminary model runs (both for ), respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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Figure 7.5. Final computational mesh; see Figure 7.1(a) for its domain extent in relation to the boundary of the second generation of the finite element mesh . . . . . . . . . . . . . . 121

Figure 7.6. (a,c) Vectors and (b,d) magnitudes (cm/s) of the residual circulation occurring through Jupiter Inlet and the north arm of the AIW, as based on the applications of the second generation and final version of the finite element mesh (both for

), respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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Figure A.1. Two-dimensional geometry of the Earth-Moon gravitational system . . . . . . . . . 134

Figure A.2. (a) Vertical tidal forces, which are greatest at the equator, zero at 35° latitude, and reversed at the poles, and (b) horizontal tidal forces, which are greatest at 45° latitude (after Pugh [2004]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure A.3. Three-dimensional geometry of the Earth-Moon gravitational system . . . . . . . . 137

Figure A.4. Exaggerated equilibrium tidal ellipsoid for a water-covered Earth where the dashed line represents the equilibrium surface under no tidal forces and the solid line represents the equilibrium surface under tidal forces (after Knauss [1978]) . . . . 138

xiv

Figure B.1. Standard deviation in the sea level variations observed at Newlyn, United Kingdom, indicating the presence of the 18.61-year nodal modulation (after Pugh [2004]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure C.1. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Coast Guard Dock, corresponding to October 2003 . . . . . . . . . . . . . . . . . . . . . . . 143

Figure C.2. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Coast Guard Dock, corresponding to November 2003 . . . . . . . . . . . . . . . . . . . . . 144

Figure C.3. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Coast Guard Dock, corresponding to December 2003 . . . . . . . . . . . . . . . . . . . . . 145

Figure C.4. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Coast Guard Dock, corresponding to January 2004 . . . . . . . . . . . . . . . . . . . . . . . 146

Figure C.5. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Coast Guard Dock, corresponding to February 2004 . . . . . . . . . . . . . . . . . . . . . . 147

Figure C.6. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Coast Guard Dock, corresponding to March 2004 . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure C.7. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at

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Coast Guard Dock, corresponding to April 2004 . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure C.8. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to October 2003 . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure C.9. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to November 2003 . . . . . . . . . . . . . . . . . . . . . . . 151

Figure C.10. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to December 2003 . . . . . . . . . . . . . . . . . . . . . . . 152

Figure C.11. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to January 2004 . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure C.12. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to February 2004 . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure C.13. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to March 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure C.14. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Pompano Drive, corresponding to April 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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Figure C.15. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to October 2003 . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure C.16. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to November 2003 . . . . . . . . . . . . . . . . . . . . . . 158

Figure C.17. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to December 2003 . . . . . . . . . . . . . . . . . . . . . . . 159

Figure C.18. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to January 2004 . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure C.19. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to February 2004 . . . . . . . . . . . . . . . . . . . . . . . . 161

Figure C.20. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to March 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure C.21. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Boy Scout Dock, corresponding to April 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure C.22. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at

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Kitching Creek, corresponding to October 2003 . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure C.23. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Kitching Creek, corresponding to November 2003 . . . . . . . . . . . . . . . . . . . . . . . 165

Figure C.24. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Kitching Creek, corresponding to December 2003 . . . . . . . . . . . . . . . . . . . . . . . . 166

Figure C.25. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Kitching Creek, corresponding to January 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Figure C.26. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Kitching Creek, corresponding to February 2004 . . . . . . . . . . . . . . . . . . . . . . . . . 168

Figure C.27. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Kitching Creek, corresponding to March 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Figure C.28. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at Kitching Creek, corresponding to April 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Figure C.29. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to October 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . 171

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Figure C.30. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to November 2003 . . . . . . . . . . . . . . . . . . . . . . . . 172

Figure C.31. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to December 2003 . . . . . . . . . . . . . . . . . . . . . . . . 173

Figure C.32. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to January 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Figure C.33. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to February 2004 . . . . . . . . . . . . . . . . . . . . . . . . . 175

Figure C.34. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to March 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Figure C.35. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red solid line) for the water level gaging station located at River Mile 9.1, corresponding to April 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Figure E.1. Computed meteorological residuals (blue solid line) plotted against the resynthesized seasonal variation (red solid line), corresponding to the water level gaging station at Coast Guard Dock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Figure E.2. Computed meteorological residuals (blue solid line) plotted against the resynthesized seasonal variation (red solid line), corresponding to the water level

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gaging station at Pompano Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Figure E.3. Computed meteorological residuals (blue solid line) plotted against the resynthesized seasonal variation (red solid line), corresponding to the water level gaging station at Boy Scout Dock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Figure E.4. Computed meteorological residuals (blue solid line) plotted against the resynthesized seasonal variation (red solid line), corresponding to the water level gaging station at Kitching Creek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Figure E.5. Computed meteorological residuals (limited by the amount of historical water level data available; blue solid line) plotted against the resynthesized seasonal variation (red solid line), corresponding to the water level gaging station located at River Mile 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

xx

ABBREVIATIONS

The following abbreviations are used in this thesis:

ACR Atlantic Coastal Ridge

ADCIRC-2DDI Advanced Circulation Model for Oceanic, Coastal, and Estuarine Waters (Two-Dimensional, Depth-Integrated Option)

AIW Atlantic Intracoastal Waterway

ATC Average Tidal Cycle

BG British Gravitational

CH3D Three-Dimensional Model of Curvilinear Hydrodynamics

CP Carte Parallelogrammatique

CWMA Corbett Wildlife Management Area

FDEP Florida Department of Environmental Protection

GWCE Generalized Wave Continuity Equation

JDSP Jonathon Dickinson State Park

JID Jupiter Inlet District

LTEA Localized Truncation Error Analysis

MLLW Mean Lower Low Water

MSD Mean Solar Day

MSL Mean Sea Level

RMS Root Mean Square

SFWMD South Florida Water Management District

SI Systeme Internationale d’Unites

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SIMSYS-2D Two-Dimensional, Estuarine-Simulation System

SMS Surface-Water Modeling System

USGS United States Geological Survey

WNAT Western North Atlantic Tidal

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CONVERSION FACTORS AND PHYSICAL CONSTANTS

All quantities presented herein are expressed in the Systeme Internationale d’Unites (SI)

system of measurement. The following conversion factors, as taken from Zwillinger (2003),

may be used to convert from the SI system of measurement to the British Gravitational (BG)

system of measurement:

Multiply SI units By To obtain BG units

centimeters (cm) 0.393701 inches (in)

cubic meters (m3) 1.307951 cubic yards (cy)

cubic meters per second (cms) 35.314670 cubic feet per second (cfs)

kilometers (km) 0.621371 miles (mi)

meters (m) 3.280840 feet (ft)

radians (rad) 57.295780 degrees (°)

square kilometers (km2) 247.105397 acres (ac)

square kilometers (km2) 0.386102 square miles (mi2)

where temperature conversions follow ( )3295

−= FC θθ and Cθ and Fθ are the temperatures in

degrees Celsius and Fahrenheit, respectively. The following linear (nautical) measurements may

aid in converting between geophysical (spherical) and Cartesian space (Zwillinger, 2003): 1° of

latitude 111.0 km; 1° of longitude at 40° latitude ≈ ≈ 85.3 km. The following physical

constants are included in this thesis (Zwillinger, 2003): G (gravitational constant) ≈

cm( ) 810003.0673.6 −×± 3/g s2; g (acceleration due to gravity, MSL at 45° latitude) ≈ 9.806194

m/s2.

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DATUM TRANSFORMATIONS

All tidal elevations presented herein are expressed in quantities of length as measured

from mean sea level (MSL). The following vertical tidal datums, as taken from the Center for

Operational Oceanographic Products and Services, Published Benchmark Sheet for the

Loxahatchee River, Florida (http://140.90.121.76/benchmarks/8722481.html; website accessed

on September 6, 2005), may serve useful in converting from MSL to another reference of

measure. Note that all vertical tidal datums listed below are referenced from mean lower low

water (MLLW).

Vertical tidal datum Elevation above MLLW (m)

Mean higher high water 0.680

Mean high water 0.635

North American vertical datum 1988 0.635

Mean tide level 0.314

MSL 0.340

Mean low water 0.047

MLLW 0.000

xxiv

NOTATION

The following symbols are used in this thesis:

jljkA , = element of the interaction matrix resulting from the interference of a satellite with the main tidal constituent;

a = offshore amplitude of the M2 tidal constituent;

C = hour angle of the Moon;

CC = Chezy friction coefficient;

fC = bottom friction factor;

minfC = minimum bottom friction factor that is approached in deep waters when the hybrid bottom friction formulation reverts to a standard quadratic bottom friction function;

#C = Courant number;

c = speed of a traveling wave in shallow water;

D = depth of the vertical water column;

ld = declination of the Moon;

2hE = horizontal eddy viscosity;

F = mutual force of attraction between two self-attracting particles;

FF = form factor;

f = Coriolis parameter;

DWf = Darcy-Weisbach friction factor;

nf = tidal constituent nodal factor;

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G = universal gravitational constant;

nG = tidal constituent phase lag on the equilibrium tide phase at the Prime Meridian;

g = acceleration due to gravity;

ng = tidal constituent phase lag relative to some defined time zero;

H = total height of the vertical water column;

breakH = break depth to determine if the hybrid bottom friction formulation will behave as a standard quadratic bottom friction function or increase with water depth similar to a Manning’s type bottom friction function;

nH = tidal constituent amplitude;

Hist = time-averaged historical tidal elevation;

ampHist = average amplitude of the historical tidal signal;

iHist = time-dependent historical tidal elevation;

h = bathymetric depth, relative to MSL;

h = mean estuarine channel depth;

i = time index;

fai − = Doodson numbers;

L = wavelength of a traveling wave;

( )φjL = latitude- and tidal species-dependent functions of the Newtonian equilibrium tide potential;

λM = depth-integrated momentum dispersion in the longitudinal direction;

φM = depth-integrated momentum dispersion in the latitudinal direction;

iMod = time-dependent model tidal elevation;

m = mass of a particle;

em = mass of the Earth;

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lm = mass of the Moon;

N = total number of terms to include in a summation;

n = tidal constituent index;

Mn = Manning’s friction factor;

( )tO = time-series observed tidal elevations;

( )xPn = Legengre polynomials of order n for variable x ;

Sp = atmospheric pressure at the free surface;

R = radius of the Earth;

2R = coefficient of determination;

RAY = Rayleigh criterion factor;

RMS = normalized RMS error;

r = distance of separation between two self-attracting particles;

jljkr , = ratio of the equilibrium amplitudes of the satellite tidal constituents to those of the major contributors;

lr = distance of separation between the Earth and Moon;

( )tS = time-series meteorological residual;

( )tT = time-series resynthesized tidal elevations;

nT = tidal constituent period;

spanT = time span of a tidal record to be analyzed;

t = time;

0t = reference time;

U = depth-integrated velocity in the longitudinal direction;

nu = tidal constituent equilibrium argument;

V = depth-integrated velocity in the latitudinal direction;

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cV = volume of water contained in channels at MSL;

nV = equilibrium tidal constituent phase lag relative to some defined time zero;

sV = volume of water stored between mean high and low water in tidal flats and marshes;

x = longitudinal axis of the estuary;

x′ = longitudinal component of horizontal (CP) space;

y ′ = latitudinal component of horizontal (CP) space;

0Z = tidal resynthesis term representative of local MSL;

α = effective Earth elasticity factor;

jljk ,α = phase corrections for the satellite tidal constituents;

γ = dimensionless parameter that describes how quickly the bottom friction factor increases as water depth decreases;

1Δ = astronomical constant involving the masses and distances associated with a celestial system;

jklΔ = phase difference between the satellite tidal constituents and the major contributors;

tΔ = time step;

xΔ = nodal spacing;

ζ = free surface elevation, relative to MSL;

η = Newtonian equilibrium tide potential;

θ = dimensionless parameter that establishes how rapidly the bottom friction factor approaches its upper and lower limits;

Cθ = temperature in degrees Celsius;

Fθ = temperature in degrees Fahrenheit;

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λ = degrees longitude (east of Greenwich positive);

0λ = longitudinal center of the CP projection;

0ρ = reference density of water;

nσ = tidal constituent frequency;

0τ = GWCE weighting parameter;

λτ S = applied free surface stress in the longitudinal direction;

φτ S = applied free surface stress in the latitudinal direction;

∗τ = quadratic bottom stress;

φ = degrees latitude (north of equator positive);

0φ = latitudinal center of the CP projection;

ϕ = absolute average phase error;

nϕ = tidal constituent phase lag relative to some defined time zero;

Ω = angular speed of the Earth;

PΩ = gravitational potential at a point P on the Earth’s surface;

nω = tidal constituent angular speed.

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CHAPTER 1. INTRODUCTION

The Loxahatchee River estuary, located on the east coast of Florida within northern Palm Beach

and southern Martin counties, empties into the Atlantic Ocean through Jupiter Inlet (Figure 1.1).

The estuarine system is comprised of three major tributaries: the Northwest Fork (Loxahatchee

River); the North Fork; the Southwest Fork. The Atlantic Intracoastal Waterway (AIW) runs

parallel to the coastline and intersects the Loxahatchee River between the central embayment and

Jupiter Inlet.

Human activities have altered the natural drainage patterns occurring within the

Loxahatchee River estuary. Prior to development, nearly level, poorly drained lands, which were

subject to frequent flooding, characterized most of the watershed region. As a result, a primary

and several secondary drainage systems and associated water-control facilities were constructed

in order to transform the Loxahatchee River watershed into an area suitable for agricultural and

residential development. Some notable structural changes that are considered here include

excavation and stabilization of Jupiter Inlet, dredging, filling, and bulkheading within the estuary

and along the Loxahatchee River, and the construction of major canals and water-control

structures. Over a century of water-control and structural modifications made to this estuarine

system has led to changes in the quality, quantity, timing, and distribution of surface water

inflows delivered to the Loxahatchee River estuary, in addition to lowering the groundwater

table within the surrounding watershed (McPherson and Sabanskas, 1980).

1

Figure 1.1. Map of the Loxahatchee River estuary, including the locations of the five water level gaging stations (Coast Guard

Dock, Pompano Drive, Boy Scout Dock, Kitching Creek, and River Mile 9.1, corresponding to the circles numbered

1-5, respectively) situated within its interior.

2

Coastal development has also greatly affected the hydrology of the Loxahatchee River

estuary. Historical evidence indicates that the mouth of the estuary, Jupiter Inlet, has been

opened and closed many times in the past as the result of natural causes. Originally, the inlet

was maintained open by surface water inflows supplied not only by the Loxahatchee River, but

also from Lake Worth Creek and Jupiter Sound, as located in the north and south arms of the

AIW, respectively. (Refer to Figure 1.1 for a map of the Loxahatchee River estuary which

highlights these two regions of the AIW.) Near the turn of the century, some of these surface

water inflows were diverted by the creation of the AIW and Lake Worth Inlet and by the

modification of St. Lucie Inlet (Vines, 1970). Subsequently, Jupiter Inlet remained closed much

of the time until 1947, except when periodically dredged. Since 1947, the inlet has been kept

open to the sea through regular dredging (McPherson et al., 1982).

As a consequence of these drainage-basin alterations, inlet modifications, and dredging

activities, groundwater levels within the adjacent floodplains have been lowered and freshwater

river inflows feeding the estuary have been reduced or altered in direction or period of flow

(McPherson and Sabanskas, 1980). This has led to the upstream migration of saltwater into the

historical freshwater reaches of the Loxahatchee River, which is the likely cause of altered

floodplain cypress forest communities found along the Northwest Fork and some of its

tributaries. Mangroves are replacing cypress forest and areas of mixed swamp hardwoods have

reacted to different degrees to the saltwater stresses. Russell and McPherson (1984) conducted

an intensive study to investigate the relationship between salinity distribution and freshwater

river inflow in the Loxahatchee River estuary, using tidal, salinity, and river-discharge data

corresponding to the dates between 1980 and 1982. More recently, studies conducted by Dent

and Ridler (1997) indicate that freshwater river inflows delivered to the Northwest Fork are

3

insufficient to maintain freshwater conditions in the Loxahatchee River around the watershed

areas affected by saltwater intrusion.

To this end, the South Florida Water Management District (SFWMD), in cooperation

with the Florida Department of Environmental Protection (FDEP), as part of a research effort to

establish minimum flows and levels for the Loxahatchee River, developed a two-dimensional,

hydrodynamic/salinity model for the estuary (SFWMD, 2002). The purpose of this modeling

effort was to provide predictions of the salinity expected at various locations within the estuary

with respect to freshwater river inflows and tidal fluctuations (Hu, 2002). Since this estuary

model did not include a groundwater component, it could not answer questions related to

saltwater intrusion and the associated effects on the vegetation within the surrounding watershed.

Hence, an integrated, surface/groundwater, three-dimensional model has been developed to

simulate river and estuarine hydrodynamics and salt transport in both surface water and

groundwater for the Loxahatchee River estuary. It is the purpose of the SFWMD to implement

this integrated, three-dimensional estuary model in order to provide salinity predictions within

the Loxahatchee River and vegetation root zone of the adjacent floodplains. As a result,

saltwater intrusion on the Northwest Fork and the feasibility of a saltwater barrier on the

Loxahatchee River will be more thoroughly investigated.

The primary focus of the present study concentrates on generating nearshore, tidal

elevation data which will be used to force the open-ocean boundary of the integrated, three-

dimensional estuary model. A large-scale computational domain that describes the western

North Atlantic Ocean, Gulf of Mexico, and Caribbean Sea is extended to include the

Loxahatchee River estuary and a limited portion of the AIW. This initial version of the finite

element mesh is applied in preliminary tidal simulations, using a two-dimensional, depth-

4

integrated, finite element-based code for coastal and ocean circulation, ADCIRC-2DDI, for

computations. A statistical analysis of the errors between model output and historical data at five

locations within the Loxahatchee River estuary (see Figure 1.1) provides absolute average phase

errors and goodness-of-fit measures that indicate a need for improvement.

Model calibration then follows with adjustments in bottom friction parameterization and

the application of (advective) freshwater river inflows; however, this sensitivity analysis fails to

improve the model response within the Loxahatchee River estuary to within acceptable levels.

Therefore, a second generation of the finite element mesh is produced in order to extend the

AIW to the north and south from the current domain extent, and to include the description of Fort

Pierce and St. Lucie Inlets and Lake Worth Inlet to the north and south, respectively, of Jupiter

Inlet. Tidal simulations follow and computed phase and amplitude errors highlight the

importance of including the AIW in the computational domain.

Finally, globally computed velocity residuals reveal a significant net circulation within

the north arm of the AIW in relation to the weak patterns in net mass transport observed through

the south arm of the AIW. A final version of the finite element mesh is then produced by

truncating the north and south arms of the AIW at a reasonable distance from Jupiter Inlet,

whereby reasonable refers to providing enough spatial coverage of the AIW to accurately

reproduce the circulation patterns within the Loxahatchee River estuary without excessively

increasing the computational requirement of the integrated, three-dimensional estuary model.

5

CHAPTER 2. TIDAL ANALYSIS

It is assumed that in the following discussion, a general knowledge of the tides is understood;

however, to facilitate this review on tidal analysis, the works of Darwin (1911), Doodson (1921),

Schureman (1941), Cartwright and Taylor (1971), Cartwright and Edden (1973), Knauss (1978),

Schwiderski (1980), Pugh (1987), Reid (1990), Deacon (1997), Cartwright (1999), Open

University (2000), and Pugh (2004) may be referenced to provide a thorough account of the

equilibrium tides. It is noted, however, that while equilibrium tidal theory provides insight into

the instantaneous response of the sea surface due to the tide-generating forces, disagreement

exists between the equilibrium tides and observed tidal heights. These discrepancies are due to

the incomplete description of the tides as offered by equilibrium tidal theory alone. Thus, a

dynamic theory of the tides which recognizes the relationship between the periodic external

forces and the natural frequencies and frictional characteristics of the interconnected ocean

basins was established. More detailed explanations regarding dynamical oceanography and real

ocean tides can be found in Darwin (1911), Proudman (1953), Defant (1960), Dietrich and Kalle

(1963), McLellan (1965), Macmillan (1966), Neumann and Pierson (1966), Phillips (1966),

Pickard (1975), LeBlond and Mysak (1978), Schwiderski (1980), and Reid (1990).

As a brief review of the various tides that are observed on Earth, the dominant periodic

geophysical forcing is the variation of the gravitational field as exerted on the Earth’s surface

and as caused by the recurring motions of the Earth-Moon and Earth-Sun systems. (Refer to

Appendix A for an outline of the formal mathematical development of gravitational forces and

the equilibrium tide as based on potential theory.) Movements due to these astronomically

6

induced gravitational forces are called either gravitational or, more usually, astronomical tides.

Further, these gravitational body forces act directly on deep oceanic waters. Tidal effects in

coastal regions, however, are not directly forced by these astronomically induced gravitational

forces, and as a result, tides near the coast arise as a side effect of deep oceanic variability,

propagating through shallower coastal waters as a wave or a combination of waves. There are

also much smaller movements due to regular meteorological forces; these are called either

meteorological or, more usually, radiational tides.

Tidal analysis, in the most basic sense, is a special case of time-series study; the idea is to

condense a long-term record of observations into a brief collection of time-invariant constants.

Due to the regularity of the tide-generating forces (e.g., those resulting from the relative [to

Earth] motions of the Moon and Sun), periodicities contained within a tidal record may be

extracted in order to describe the tidal displacement at a location as a sum of the associated

harmonics. For a historical review, various methods of such harmonic analyses, as devised by

Darwin (1911), Doodson (1928), and Horn (1960), are primarily aimed at determining the

amplitude and phase properties of the predominant harmonics. More recently, attempts have

been made to evaluate the contribution of non-tidal phenomena present in the record of

observations in order to provide a quantitative estimate of the variability in the tidal record

(Munk and Cartwright, 1966). The following section on tidal analysis covers a brief review of

the mathematics involved with the analysis of the tides, a discussion regarding harmonic

constants and their role in representing the tides, and an example harmonic analysis procedure, as

applied to the historical water level data that are used in the present study.

Specialized techniques have been devised to take advantage of the deterministic nature of

the tides. In classical harmonic analysis, the tidal forcing is modeled as a set of spectral lines,

7

and hence, Fourier series forms the basis of the harmonic analysis of the tides; a superposition of

multiple sinusoidal waves, each with its own properties (e.g., interval of recurrence and those

associated with the amplitude and phase of the tidal component), to form a total tidal signal.

Therefore, tidal variations can be represented by a finite number N of harmonic terms of the form

(Cartwright and Taylor, 1971):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.1) ( )nnn gtH −ωcos

where n = component index; Hn = component amplitude; ωn = component angular speed = 2π/Tn;

Tn = component period; gn = component phase lag relative to some defined time zero (commonly

taken as the phase lag on the equilibrium tide phase at the Prime Meridian, in which case it is

called Gn); t = time.

Due to the nearly linear nature of the dynamics between the tide-generating forces and

the associated ocean response, it is implied then that the forced response of the ocean surface

contains only those frequencies present in the tide-generating forces. Hence, use of the

equilibrium tide is helpful in determining the angular speeds of the various tidal components.

These are found by an expansion of the equilibrium tide into harmonic terms; the speeds of these

terms are found to have the general form (Doodson, 1921):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.2) ( )terms,, 654321 ωωωωωωω +++= cban iii

where the values of ω1 to ω6 are the angular speeds related to the astronomical parameters listed

in Table 2.1 and the coefficients ia to ic are small integers, usually in the range between -2 and 2.

8

Thus, a specific set of these six integers (referred to as the Doodson numbers) may be applied

(through Eq. [2.2]) to the fundamental frequencies listed in Table 2.1 in order to specify a

particular tidal frequency (Godin, 1972).

Table 2.1. The basic speeds and origins of the astronomical arguments that give the

frequencies of the harmonic components (after Harris [1991]).

Origin Period Degrees per mean solar hour Symbol

Mean solar day (MSD) 1.0000 MSD 15.0000 ω0

Mean lunar day 1.0351 MSD 14.4921 ω1

Sidereal month 27.3217 MSD 0.5490 ω2

Tropical year 365.2422 MSD 0.0411 ω3

Moon's perigee 8.85 years 0.0046 ω4

Regression of Moon's nodesa 18.61 years 0.0022 ω5

Perihelion 20942 years – ω6

a Refer to Appendix B for an overview of nodal cycles.

At this point in the harmonic analysis, the individual harmonic components (herein

referred to as tidal constituents) are derived by considering the associated periodicities of the

corresponding tide-generating forces. For example, the M2 tidal constituent is representative of

the semi-diurnal (with a period of 12 hours and 25 minutes) tide resulting from the Moon’s

revolution about the Earth in a circular orbit. The naming convention follows that the letter M

represents the Moon and the number 2 indicates that the tide occurs twice a day. Similarly, the

semi-diurnal tide generated by the Sun (as being on the equatorial plane of the Earth) has a

9

period of exactly 12 hours, and hence, the S2 tidal constituent is represented. It is noted here that

the combination of these two tides (M2, S2) produces the spring-neap tidal cycle (Figure 2.1).

Figure 2.1. (a) Spring tide conditions when the Moon is in syzygy and (b) neap tide conditions

when the Moon is in quadrature (after Pugh [2004]).

These concepts are now related to the actual movements of the Moon and Sun by

considering each individual modulation (e.g., those associated with the Moon’s phase, distance

from Earth, and declination) as an effect produced by a separate phantom satellite (Pugh, 2004).

For instance, the astronomical expressions can be expanded for the Moon’s phase, distance from

Earth, and declination mathematically to determine the periods and theoretical amplitudes of the

extra terms. The concept is then extended to include the longer-period variations of the Moon

and Sun, which results in annual, semi-annual, and diurnal tidal constituents. When this full

10

expansion of the equilibrium tide is done for all modulations associated with the Moon and Sun,

the resulting list of tidal constituents may be very long. Nevertheless, examination of the relative

amplitudes of the tidal constituents arising from the mathematical expansion of the equilibrium

tide shows that only a few harmonics are dominant (Table 2.2).

Table 2.2. The dominant harmonics of the tides and their physical causes (after Reid [1990]).

ib icPeriod (MSD)

Degrees per solar hour

Equilibrium amplitude(M2 = 1.0000) Origin

Long-period ia = 0

SA 0 1 364.96 0.0411 0.0127 Solar annual

SSA 0 2 182.70 0.0821 0.0802 Solar semi-annual

Diurnal ia = 1

Q1 -2 0 1.120 13.3987 0.0795 Lunar ellipse

O1 -1 0 1.076 13.9430 0.4151 Principal lunar

P1 1 -2 1.003 14.9589 0.1932 Principal solar

K1 1 0 0.997 15.0411 0.5838 Principal lunar and solar

Semi-diurnal ia = 2

N2 -1 0 0.527 28.4397 0.1915 Lunar ellipse

M2 0 0 0.518 28.9841 1.0000 Principal lunar

L2 1 0 0.508 29.5285 0.0238 Lunar ellipse

S2 2 -2 0.500 30.0000 0.4652 Principal solar

K2 2 0 0.499 30.0821 0.1266 Declinational lunar and solar

The line spectra of the diurnal and semi-diurnal tidal constituents are plotted in Figure 2.2,

which shows the frequencies of the terms in the fuller expansion of the equilibrium tide and

confirms the significance of the dominant harmonics. The frequency-dependent pattern of tidal

constituents shown in Figure 2.2 can be explained in terms of Eq. (2.2). The main divisions in

the pattern of tidal constituents are the number of cycles per day (governed by ia), where each

11

division is called a tidal species. In the complete astronomical expansion, ib is used to fit the

monthly modulations, which varies between -5 and 5 and defines the group within each tidal

species. Within each group, ic fits the annual modulations; it also varies between -5 and 5 and is

said to define the tidal constituent.

Modulations in ω4, ω5, and ω6 (see Eq. [2.2]) are affected by longer-period astronomical

cycles and cannot be resolved as independent harmonics from a year of observations (see

Appendix B). Therefore, variations in these astronomical arguments are represented in the

harmonic expansions by small adjustment factors to the amplitude and phase. These nodal

adjustment factors, fn (nodal factor) and un (equilibrium argument), are applied individually to

the lunar tidal constituents through Eqs. (2.1) and (2.2) in order to account for the long-term

nodal modulations (Cartwright and Taylor, 1971):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.3) ( )[ ]nnnnn ugtfH +−ωcos

It is noted that the nodal factor and equilibrium argument are set to 1.0 and 0.0, respectively, for

the solar tidal constituents, as there are no nodal effects on the solar-induced tides.

12

Figure 2.2. Frequency-dependent pattern of the (a) diurnal and (b) semi-diurnal tidal

constituents with their associated equilibrium amplitudes plotted on a logarithmic

scale (after Cartwright and Edden [1973]). Each individual vertical line represents

a tidal constituent; note the clustering of tidal constituents into groups within each

tidal species.

13

In applying the harmonic method of analysis to a tidal record, a tidal function T(t) is fit to

sea level observations (Godin, 1991b):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.4) ( ) ( )[ ]∑ ++−+=N

nnnnnn uVgtfHZtT ωcos0

where the unknown parameters are Z0 and the series of tidal constituent amplitudes and phases

(Hn, gn). Z0 is included here as a variable to be fitted in the analysis, but it commonly represents

local mean sea level (MSL) and is therefore a known parameter. The nodal adjustment factors

are given as fn and un and the terms ωnt and Vn together determine the phase angle of the

equilibrium tidal constituent. Vn is the equilibrium phase angle for the tidal constituent at the

arbitrary time origin. The accepted convention is to take Vn as for the Prime Meridian and t in

the standard time zone of the observation station.

A least-squares fitting procedure is then employed to determine the amplitudes and

phases of the tidal constituents corresponding to the particular measurement site. This least-

squares fitting procedure serves to minimize ( )∑ tS 2 , the square of the residual differences

between the observed O(t) and computed tidal elevations when summed over all observations

(Godin, 1991b):

( ) ( ) ( )tTtOtS −= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.5)

The computational aspects of the least-squares fitting procedure involve matrix algebra

and go beyond the scope of this review on tidal analysis; however, Foreman (1977) gives a

thorough account of the problem formulation and matrix solution as related to fitting a tidal

14

function to sea level observations. It is noted, however, that the system of equations to be solved

may be written schematically as follows: (observations [known]) = (equilibrium tide [known]) ×

(empirical constants [unknown]). Moreover, Pugh (2004) remarks on the following useful

properties that the least-squares fitting procedure offers: gaps in the data are permissible; any

length of data may be treated (usually complete months or years are analyzed); no assumptions

are made about data outside of the interval to which the fit is made; transient phenomena are

eliminated (i.e., only variations with a coherent phase at tidal frequencies are extracted); any

computational time step may be employed in the analysis albeit fitting if often applied to hourly

values.

There are certain rules for deciding which harmonic amplitudes and phases are to be

determined from a tidal analysis. In general, the longer the length of the data record involved in

the tidal analysis, the greater the number of tidal constituents may be extracted. Selection of the

tidal constituents to include in the tidal analysis is often governed by the Rayleigh criterion,

which requires that only harmonics separated by at least a complete period from their

neighboring harmonics over the length of data available be included in the tidal analysis. For

example, consider the frequencies of two individual tidal constituents, 0σ and 1σ , and the time

span Tspan of the data record, to be analyzed. For both tidal constituents to be included in the

tidal analysis, the Rayleigh criterion must be satisfied (Foreman, 1977):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.6) RAYTspan ≥− 10 σσ

where RAY is commonly specified to be equal to unity.

15

Presented in an alternative way, to determine the M2 and S2 tidal constituents (with

angular speeds of 28.9841 and 30.0000 degrees per hour, respectively; see Table 2.2)

independently in a tidal analysis requires a data record of the following minimum length be used:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.7) ( ) days77.14hr9841.280000.30

360=

− o

o

where this minimum length of the data record required to resolve a pair of tidal constituents is

known as the synodic period (Pugh, 2004). It is noted that in the previous case, the synodic

period required to separate the M2 and S2 tidal constituents is equal to the recurrence interval of

the spring-neap tidal cycle (see Figure 2.1).

For general use, an automated selection algorithm devised by Foreman (1977) is

currently in place, which works as follows. First, all possible tidal constituents are gathered and

listed in order of decreasing equilibrium amplitude (see Table 2.2). Less important tidal

constituents (e.g., those with lesser equilibrium amplitudes) whose frequencies are less than a

Rayleigh resolution limit (see Eq. [2.6]) apart from more important tidal constituents (e.g., those

with greater equilibrium amplitudes) are then discarded. Finally, additional tidal constituents

may be explicitly added to the list, if required.

Before continuing on with the tidal analysis, it is also necessary to satisfy another basic

rule of time-series analysis, as related to the frequency at which observations are made. The

Nyquist criterion states that only terms having a period longer than twice the sampling interval

can be resolved. In the usual case of hourly data sampling, this shortest period is two hours, so

that resolution of the twelfth-diurnal (with a period of 2 hours and 4 minutes) M12 tidal

constituent would just be possible. In practice, however, this is not a severe restriction except in

16

very shallow waters, where sampling more frequently than once an hour is necessary to represent

these shallow-water tides.

It may be discovered that due to the limiting length of the data record, many of the

possible harmonics to include in the tidal analysis are not resolvable, as restricted by the

Rayleigh criterion. The standard approach that is then taken to deal with this issue is to form

clusters containing all of the tidal constituents with the same first three Doodson numbers (see

Eq. [2.2]). The major contributor (e.g., the tidal constituent with the greatest equilibrium

amplitude) lends its name to the cluster and the remaining contributors are called satellite tidal

constituents. The tidal analysis, using these main and satellite tidal constituents, then continues

(as described by Foreman [1977]) in the following manner. The Rayleigh criterion is applied to

the frequencies of the main tidal constituents to determine their inclusion in or omission from the

tidal analysis. A least-squares fit is made between the tidal function (using only the main tidal

constituents) and sea level observations to obtain the apparent amplitudes and phases; however,

since these results are due to the cumulative effect of all of the tidal constituents included in the

clusters, an adjustment must be made to determine the contributions due to the main tidal

constituents alone. In order to make these nodal modulation corrections (see Eq. [2.3]) to the

main tidal constituents, it is necessary to know the relative amplitudes and phases of the satellite

tidal constituents contained within the respective clusters. As is commonly done, it is assumed

that the same relationship that is found with the equilibrium tide holds for the actual tide (i.e., the

equilibrium amplitude ratio of a satellite to its main tidal constituent is assumed to be equal to

the actual amplitude ratio, and the difference in equilibrium phase between a satellite and its

main tidal constituent is assumed to be equal to the actual phase difference).

17

Due to the presence of satellite tidal constituents in a given cluster, it is known from

equilibrium tidal theory that the analyzed signal found at the frequency of the main tidal

constituent jσ actually results from:

( ) ( ) ( )∑∑ −+−+−l

jljljljlk

jkjkjkjkjjj gVHAgVHAgVH cossinsin . . . . . . . . . . . . . . . . . (2.8)

for the diurnal and terdiurnal (occurring three times a day) tidal constituents, and:

( ) ( ) ( )∑∑ −+−+−l

jljljljlk

jkjkjkjkjjj gVHAgVHAgVH sincoscos . . . . . . . . . . . . . . . . . (2.9)

for the annual, semi-annual, and semi-diurnal tidal constituents (Cartwright and Taylor, 1971).

The single j subscripts refer to the main tidal constituents while the multiple jk and jl subscripts

refer to the satellite tidal constituents originating from the second- and third-order terms of the

tidal potential, respectively (see Appendix A). Ajk,jl is the element of the interaction matrix

resulting from the interference of a satellite with the main tidal constituent (Foreman, 1977).

It is the convention in tidal analysis, and an assumption made in the least-squares fitting

procedure, that all tidal constituents arise through a cosine term with positive amplitude;

however, the diurnal and terdiurnal tidal constituents, assuming that they are due to second-order

terms in the tidal potential, actually arise through a sine term with a (possible) negative

amplitude. Hence, a phase correction of either 41

− or 43

− cycles is necessary:

18

( )

0for43cos

0for41cossin

<⎟⎠⎞

⎜⎝⎛ −−=

≥⎟⎠⎞

⎜⎝⎛ −−=−

jjjj

jjjjjjj

HgVH

HgVHgV H

. . . . . . . . . . . . . . . . . . . . . . . . (2.10)

A similar adjustment of 21 cycle is necessary for the annual, semi-annual, and semi-diurnal tidal

constituents (only if the amplitude is negative).

Making these changes, the cluster contribution in the diurnal and terdiurnal cases is:

( ) ( ) ( )∑∑ −+′+−+′+−′l

jljljljljlk

jkjkjkjkjkjjj gVHAgVHAgVH αα coscoscos . . . . . . . . (2.11)

where if , then 0<jH 43

−=′ VV , 21

=jkα , and 43

=jlα , and if , then 0>jH 41

−=′ VV ,

0=jkα , and 41

=jlα . A further phase correction to the satellite tidal constituents is also

required. Replacing Hjk and Hjl with their absolute values results in the following adjustment

factors: 0=jkα if both and have the same sign, and jH jkH21

=jkα otherwise; 41

=jlα if

both and have the same sign, and jH jlH43

=jlα otherwise. Similarly, for the annual, semi-

annual, and semi-diurnal tidal constituents, the cluster contribution is written as:

( ) ( ) ( )∑∑ −+′+−+′+−′l

jljljljljlk

jkjkjkjkjkjjj gVHAgVHAgVH αα coscoscos . . . . (2.12)

19

where 21

+=′ VV if , and 0<jH VV =′ otherwise; 0=jkα if and have the same sign,

and

jH jkH

21

=jkα otherwise; 41

−=jlα if and have the same sign, and jH jlH41

=jlα otherwise.

When applying this cluster approach in a tidal analysis, it is assumed that the result found

contains a nodal correction made to the main tidal constituents: ( )jjjjj ugVHf +−′cos . For the

purpose of calculating these nodal adjustment factors corresponding to the main tidal

constituents, it is assumed that the admittance is nearly constant over the frequency range of the

associated cluster. Thus, gj = gjk = gjl, and jjkjk HHr = and jjljl HHr = are equal to the

ratios of the equilibrium amplitudes of the satellite tidal constituents to those of the major

contributors. Dropping the prime notation (on V) and grouping the second- and third-order tidal

potential terms into one summation (represented by the multiple jkl subscripts), the relationship

between the analyzed results for a main tidal constituent and the actual cluster combination is

represented by:

( ) ( ) ( ⎥⎦

⎤⎢⎣

⎡Δ+−++−=+− ∑

kjkljjkljjkljkljjjjjjjj gVrAgVHugVAf αcoscoscos . . . . (2.13) )

where . Expanding this result and observing that it holds for all Vjjkljkl VV −=Δ j(t), the

following explicit formulas are found for the nodal factor and equilibrium argument, respectively

(see Schureman [1941] and Schwiderski [1980]):

. . . . . . . . . . . . . . . . . . (2.14) ( ) ( )21

22

sincos1⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡+Δ+⎥

⎦

⎤⎢⎣

⎡+Δ+= ∑∑

kjkjkjkjk

kjkjkjkjkj rArAf αα

20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.15) ( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

+Δ+

+Δ=

∑∑

k jkjkjkjk

k jkjkjkjkj rA

rAu

αα

cos1sin

arctan

For a tidal analysis carried out over 2N + 1 consecutive observations and sampled at Δt time

intervals apart, the interaction-matrix element is given by (Foreman, 1977):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.16) ( ) ( )[ ]

( ) ( )[ ]2sin12212sin

jjk

jjkjk tN

tNA

σσσσ

−Δ+

−Δ+=

In the present study, historical water surface elevation data are obtained for the five water

level gaging stations located within the interior of the Loxahatchee River estuary (see Figure

1.1). These water level data sets contain time-series water surface elevations (sampled at 30-

minute intervals) corresponding to a two-year time period, which extends from January 1, 2003

to January 1, 2005, for these five water level gaging stations. (This two-year time period is

chosen in order to include the project time period, which extends from October 1, 2003 to May 1,

2004.) Upon preliminary examination of these water surface elevation data, a significant amount

of non-astronomical influence§ appears to be included in the overall measured signals (see

Appendix C). Thus, a harmonic analysis is performed on these water level data sets in order to

extract the regular tidal oscillations from the total observed signals. A tidal analysis tool written

in MATLAB computing language by Pawlowicz et al. (2002) is employed to accomplish this

current task. This package of routines (collectively named T_TIDE) is used to perform a

§ Non-astronomical influence refers to all non-astronomically driven physical processes which may affect coastal and oceanic water levels, including, but not limited to, temperature- and salinity-driven circulation, wind and pressure effects, and local resonant oscillations (i.e., seiches); however, within a semi-enclosed water body (e.g., an estuary), most non-astronomical influence may be attributed to meteorological effects.

21

classical tidal analysis on the historical water surface elevation data obtained at the five water

level gaging stations located within the interior of the estuary.

While the classical tidal analysis approach employed by T_TIDE is fully described in

Pawlowicz et al. (2002), the subsequent overview provides a brief summary of the procedure

followed by the tidal analysis tool. The astronomical variables associated with the magnitude of

the tidal potential (see Appendix A) are determined for a given Julian date using the formulas

given by Seidelmann (1992). The effects produced by the tide-generating forces are then

combined with the Doodson numbers (see Eq. [2.2]) to specify all possible tidal constituents.

Following, the long-period, semi-diurnal, and diurnal tidal species are grouped into clusters (see

Foreman [1977]), which are then collectively applied in the tidal analysis. Amplitude and phase

estimates of the tidal constituents are made using a least-squares fitting procedure (see Eq. [2.5])

through algorithms described by Godin (1972) and Foreman (1977). A total of 146 tidal

constituents may be chosen (according to the Rayleigh and Nyquist criteria) to include in the

tidal analysis (see Foreman [1977]): 45 astronomical in origin; 101 shallow water-based. Lastly,

phase corrections (see Eqs. [2.11] and [2.12]) and nodal adjustments (see Schureman [1941] and

Schwiderski [1980]) are applied to the cluster contributions in order to determine the individual

tidal constituents.

The historical tidal signal is then resynthesized over the project time period through the

summation of Eq. (2.4) using the T_TIDE-computed tidal constituents and corresponding nodal

adjustment factors (see Table 2.3). (Of importance, the solar annual [SA] and solar semi-annual

[SSA] tidal constituents are excluded from this tidal resynthesis for the purpose of eliminating

any seasonal variations within the resynthesized historical tidal signal.) Discrepancies between

the historical water surface elevations and resynthesized historical tidal signals are apparent at all

22

five water level gaging stations (see Appendix C), indicating the presence of meteorology (see

footnote on page 21) in the records of observations.

Table 2.3. 68 tidal constituents and corresponding nodal adjustment factors extracted by

T_TIDE and used in the resynthesis of the historical tidal signal.

Tidal constituenta Tidal species Period

(MSD) Degrees per solar hour

Nodal factor (-)b

Equilibrium argument (rad)b

SA long-period 365.18 0.0411 1.000 5.193

SSA long-period 182.59 0.0822 1.000 1.415

MSM long-period 31.81 0.4715 1.000 1.845

MM long-period 27.55 0.5444 1.000 1.189

MSF long-period 14.77 1.0159 1.000 3.034

MF long-period 13.66 1.0980 1.000 4.450

ALP1 diurnal 1.211 12.3828 1.129 4.208

2Q1 diurnal 1.167 12.8543 1.122 6.048

SIG1 diurnal 1.160 12.9271 1.132 5.384

Q1 diurnal 1.120 13.3987 1.126 0.953

RHO1 diurnal 1.113 13.4715 1.159 0.264

O1 diurnal 1.076 13.9431 1.131 2.136

TAU1 diurnal 1.070 14.0252 0.837 0.437

BET1 diurnal 1.041 14.4145 1.156 0.874

NO1 diurnal 1.035 14.4967 1.104 1.619

CHI1 diurnal 1.030 14.5696 1.136 1.311

PI1 diurnal 1.006 14.9179 0.995 6.136

P1 diurnal 1.003 14.9589 0.994 5.043

S1 diurnal 1.000 15.0000 0.689 4.673

K1 diurnal 0.997 15.0411 1.080 3.216

23



Nodal factor (-)b


PSI1 diurnal 0.995 15.0821 1.012 2.220

PHI1 diurnal 0.992 15.1232 0.941 4.764

THE1 diurnal 0.967 15.5126 1.136 4.988

J1 diurnal 0.962 15.5854 1.121 4.340

SO1 diurnal 0.934 16.0570 1.132 6.243

OO1 diurnal 0.929 16.1391 1.480 0.949

UPS1 diurnal 0.899 16.6835 1.508 2.214

OQ2 semi-diurnal 0.548 27.3510 0.875 1.669

EPS2 semi-diurnal 0.547 27.4238 0.937 1.054

2N2 semi-diurnal 0.538 27.8954 0.904 2.875

MU2 semi-diurnal 0.536 27.9682 0.966 2.269

N2 semi-diurnal 0.527 28.4397 0.974 4.126

NU2 semi-diurnal 0.526 28.5126 0.969 3.471

GAM2 semi-diurnal 0.519 28.9112 1.090 2.780

H1 semi-diurnal 0.518 28.9430 0.954 3.306

M2 semi-diurnal 0.518 28.9841 0.976 5.316

H2 semi-diurnal 0.517 29.0252 0.987 4.236

MKS2 semi-diurnal 0.516 29.0663 1.180 0.219

LDA2 semi-diurnal 0.509 29.4556 0.972 4.011

L2 semi-diurnal 0.508 29.5285 1.095 3.061

T2 semi-diurnal 0.501 29.9589 1.000 3.185

S2 semi-diurnal 0.500 30.0000 1.001 2.096

R2 semi-diurnal 0.499 30.0411 1.221 4.242

K2 semi-diurnal 0.499 30.0821 1.207 3.282

MSN2 semi-diurnal 0.491 30.5444 0.952 3.285

ETA2 semi-diurnal 0.490 30.6265 1.217 4.453

MO3 terdiurnal 0.349 42.9272 1.104 1.169

24



Nodal factor (-)b


M3 terdiurnal 0.345 43.4762 0.965 4.828

SO3 terdiurnal 0.341 43.9430 1.132 4.232

MK3 terdiurnal 0.341 44.0252 1.054 2.249

SK3 terdiurnal 0.333 45.0411 1.082 5.312

MN4 fourth-diurnal 0.261 57.4238 0.950 3.159

M4 fourth-diurnal 0.259 57.9682 0.953 4.349

SN4 fourth-diurnal 0.257 58.4397 0.975 6.223

MS4 fourth-diurnal 0.254 58.9841 0.977 1.129

MK4 fourth-diurnal 0.254 59.0662 1.178 2.315

S4 fourth-diurnal 0.250 60.0000 1.003 4.193

SK4 fourth-diurnal 0.250 60.0821 1.209 5.379

2MK5 fifth-diurnal 0.205 73.0093 1.029 1.281

2SK5 fifth-diurnal 0.200 75.0411 1.083 1.125

2MN6 sixth-diurnal 0.174 86.4080 0.925 1.002

M6 sixth-diurnal 0.173 86.9523 0.930 3.382

2MS6 sixth-diurnal 0.171 87.9682 0.954 0.161

2MK6 sixth-diurnal 0.170 88.0503 1.150 1.348

2SM6 sixth-diurnal 0.169 88.9841 0.979 3.226

MSK6 sixth-diurnal 0.168 89.0662 1.180 4.411

3MK7 seventh-diurnal 0.147 101.9934 1.005 0.314

M8 eighth-diurnal 0.129 115.9364 0.908 2.414 a Refer to Appendix D for a listing of the tidal constituent amplitudes and phases. b Nodal adjustment factors computed according to the 16th hour of November 1, 2003.

Meteorological effects (see footnote on page 21) contained within the records of

observations are computed through Eq. (2.5) in order to quantify these discrepancies between the

25

historical water surface elevations and resynthesized historical tidal signals. Additionally, the

solar annual (SA) and solar semi-annual (SSA) tidal constituents are resynthesized over the two-

year time period associated with the historical water level data in order to obtain the seasonal

variation contained within the overall measured signals. (This two-year time period is selected

for the purpose of presenting two and four complete cycles of the annual and semi-annual

seasonal variations, respectively.) Correlation between the computed meteorological residuals

and resynthesized seasonal variations suggests that the observed water levels are highly

influenced by long-term solar heating and weather effects (see footnote on page 21) (see

Appendix E).

To close this discussion on tidal analysis, various harmonic equivalents (through use of

the tidal constituents) of some non-harmonic terms are presented. A common non-harmonic

term used to describe the tides is associated with the fortnightly modulation in the semi-diurnal

tidal amplitudes, or the spring-neap tidal cycle (see Figure 2.1), which can be represented by the

combination of the principal lunar (M2) and principal solar (S2) tidal constituents:

( ) ( )2022120 2cos2cos SSMM gtHgtHZ −+−+ ωω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.17)

where time zero is at syzygy (see Figure 2.1) and the angular speeds of the M2 and S2 tidal

constituents, ω1 and ω0, respectively, can be found in Table 2.1. The maximum values of the

combined amplitudes are given by mean high water springs and mean low water springs,

respectively:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.18) ( )220 SM HHZ ++

26

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.19) ( )220 SM HHZ +−

and the minimum values of the combined amplitudes are given by mean high water neaps and

mean low water neaps, respectively:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.20) ( )220 SM HHZ −+

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.21) ( )220 SM HHZ −−

The relative importance of the diurnal and semi-diurnal tidal constituents may be

expressed in terms of the form factor, as computed by the ratio of the major diurnal and semi-

diurnal harmonic amplitudes:

22

11

SM

OK

HHHH

FF++

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.22)

In terms of the form factor, the tides may be roughly classified as semi-diurnal (FF = 0.00 –

0.25), mixed/semi-diurnal (FF = 0.25 – 1.50), mixed/diurnal (FF = 1.50 – 3.00), or diurnal (FF >

3.00). Using the amplitudes of the K1, O1, M2, and S2 tidal constituents extracted in the

harmonic analysis to compute the associated form factors, the tides in the Loxahatchee River

estuary can be classified as slightly mixed and strongly semi-diurnal (see Table 2.4). To provide

a relative basis, the form factors associated with the tides in the Western North Atlantic Tidal

(WNAT) model domain are displayed in Figure 2.3.

27

Table 2.4. Computed form factors associated with the tides in the Loxahatchee River estuary.

Tidal constituent amplitude (m)b

Water level gaging stationa

K1 O1 M2 S2 Form factor, FF (-)

Coast Guard Dock 0.060 0.050 0.323 0.047 0.298

Pompano Drive 0.064 0.051 0.321 0.045 0.314

Boy Scout Dock 0.058 0.048 0.308 0.046 0.300

Kitching Creek 0.058 0.048 0.313 0.048 0.293

River Mile 9.1 0.059 0.049 0.319 0.048 0.295 a Refer to Figure 1.1 for the locations of these five water level gaging stations. b Refer to Appendix D for a listing of the tidal constituent amplitudes and phases.

Tides have also been classified in various other general ways that can be related to the

tidal constituent amplitudes. One very crude classification of the tides that is still in use today is

given as follows: tides with a range greater than 4 m are called macrotidal; those with a range

between 2 and 4 m are called mesotidal; those with a range less than 2 m are called microtidal.

Over the project time period, the range of the tides experienced at the five water level gaging

stations located within the estuary varies between 0.50 and 1.00 m (see Appendix C). Thus, the

tides within the Loxahatchee River estuary can be further classified as being microtidal.

28

Figure 2.3. Computed form factors associated with the tides in the WNAT model domain,

highlighting the diurnal (FF 3.00) and semi-diurnal (FF = 0.00 – 0.25) tidal

regimes experienced within the Gulf of Mexico and Caribbean Sea and in the

western North Atlantic Ocean, respectively.

≥

29

CHAPTER 3. LITERATURE REVIEW

While the major focus of the research presented herein involves the analysis, modeling, and

simulation of the tides in the Loxahatchee River estuary, the following literature review serves to

cover three main topics directly related to the present study. (In addition, see Chapter 2 for a

review on tidal analysis, as those concepts and methods also relate to the tidally induced

circulation patterns occurring within the estuary.) First, recent progress in the simulation of tidal

circulation patterns using two- and three-dimensional numerical models is documented to

highlight past advancements and demonstrate the need for more advanced modeling methods.

Following, previous investigations involving the Loxahatchee River estuary are reviewed in

order to gain knowledge from past modeling efforts that dealt with studying the tides occurring

within this estuarine system. Lastly, a section dedicated to the calculation and evaluation of

residual circulation patterns offers useful information related to the analysis of net tidal flows

experienced within the Loxahatchee River estuary.

3.1. Recent Progress in the Two- and Three-dimensional Modeling of Tides

An understanding of the circulation patterns occurring within the estuary is necessary to

investigate the physical, chemical, and biological processes apparent within the water body. To

this end, considerable effort has been devoted to the study of tidal circulation patterns existent

within estuaries and other small water bodies (Lynch, 1983; Westerink and Gray, 1991). Early

work in tidal dynamics was largely confined to analytical studies using linearized versions of the

30

complete equations of motion (Lamb, 1932). With the growth of computers, however, numerical

models began to replace their analytically based predecessors. Resulting from such

technological advancement, finite difference methods were first implemented to solve the

complete equations of motion (Leendertse, 1967). Consequently, the finite difference method

was expanded to include the transport equations, which offered unique applications to estuarine

systems (Reid and Bodine, 1968; Leendertse, 1970; Leendertse and Gritton, 1971; Hess, 1976).

More recently, numerous researchers have begun an examination of the finite element

method, partly because geometric complexities, which are often characteristic of estuaries, are

better handled in this approach than by the finite difference method. Advances have been made

in finite element modeling using vertically integrated approximations to the complete equations

of motion, with strides being made using three-dimensional approaches of the finite element

method. These two-dimensional, finite element-based modeling applications have utilized a

variety of formulations and solution techniques such as the use of various basis functions,

different matrix storage systems, linear and non-linear solution methods, and distinct time-

stepping schemes (see Connor and Wang [1973], King et al. [1975], Kawahara et al. [1976],

Partridge and Brebbia [1976], Pearson and Winter [1977], Kawahara et al. [1978], and Navon

[1988]).

One common difficulty associated with the simulation of tidal circulation patterns deals

with the discretization of the temporal domain. Therefore, as an alternative to the

implementation of time-stepping schemes, frequency domain-based schemes have been shown to

provide highly efficient and stable solutions to the equations governing tidal circulation. For

example, Walters (1988) and Westerink et al. (1988) explore the harmonic solutions to the

31

vertically integrated equations of tidal motion in order examine non-linear tidal constituent

interactions in a highly controlled manner.

The propagation and recession of the wave front onto and from dry land, respectively, is

also being actively pursued through applications of the finite element method. Akanbi and

Katopodes (1988) solve the vertically integrated equations of tidal motion in their primitive form

using a moving and deforming finite element mesh, which follows the flood wave as it flows

over dry regions. A dissipative, finite element-based procedure is employed to prevent

instabilities from arising due to the highly non-linear flow regime present along the water/land

interface. Siden and Lynch (1988) solve the vertically integrated equations of tidal motion in a

generalized wave continuity equation (GWCE) formulation, which applies no dissipative devices

for stability control. In this study, a moving and deforming finite element mesh is employed to

follow the water/land interface and allow for the description of tidal flow within dry regions.

The general robustness of the GWCE formulation of the vertically integrated equations of

tidal motion has proven itself quite valuable in modeling tidal phenomena within large-scale

computational domains; however, in small water bodies where lateral viscous effects require

description, a significantly greater computational effort is required to handle the additional

lateral viscosity terms. Lynch et al. (1988) avoid this problem by including a separate equation

to describe horizontal shear stress; Kolar and Gray (1990) apply an approximation to the

primitive continuity equation and substitute a temporal derivative term for the spatial derivatives

contained in the lateral viscosity terms of the GWCE.

Continued efforts regarding the development of various discretization schemes serve to

minimize the amount of numerical noise arising in the vertically integrated solutions to the

complete equations of motion (Gray, 1982). Westerink et al. (1987) examine an equal-order

32

interpolation, finite element-based solution technique for solving the vertically integrated

equations of tidal motion in their primitive harmonic form. The improved numerical behavior of

the solution scheme is shown to avoid the generation of artificial (near 2Δx) modes that typically

plague primitive-based, finite element solutions to the vertically integrated equations of tidal

motion.

Despite the good performance of GWCE formulation-based solution techniques in

idealized, linear flow computations, field applications which are not heavily damped are still

rather wiggly (Baptista et al., 1989), indicating that other mechanisms may still excite spurious

modes in primitive-based solutions (e.g., geometric boundary irregularities, perturbations in

elevation boundary conditions, small-scale variations in bathymetry). To this end, DeVantier

(1989) presents a stream function vorticity equation formulation that is considerably more

efficient than the primitive-based solution schemes; however, significant difficulties associated

with the description of the spatial variations in eddy viscosity limit its scope of application.

Moreover, Laible (1990) applies a least-squares collocation approach using an orthogonal finite

element mesh to solve the vertically integrated equations of tidal motion, which is shown to

exhibit improved numerical amplitude and phase propagation characteristics.

Although two-dimensional, vertically integrated numerical models have progressed to the

point where many areas of agreement exist as to their proper implementation, (laterally

averaged) two- and (fully) three-dimensional modeling of tidal phenomena is not nearly as

developed due to the computational requirements needed to run such comprehensive numerical

models. To this end, there lacks sufficient unanimity towards the identification of the essential

components of the computational algorithms required to account for the vertical structure of the

tidal flow being modeled. Further requirements needed for the evaluation of such vertical-

33

structure numerical models include improved verification data sets, which may also be used to

isolate the effects of separate numerical approximations used within the respective vertical-

structure numerical models. Nonetheless, progress is being made towards improving these

vertical-structure numerical models, with emphasis placed on their capability to correctly capture

the physics occurring within the water body and to minimize the generation of spurious

numerical artifacts.

Two-dimensional numerical models which simulate vertical structure are either laterally

averaged or assume that property variation and velocity in one of the lateral directions may be

ignored. These types of numerical models have their greatest applicability in reproducing the

hydrodynamics within river systems or water bodies where one lateral coordinate may be

identified as the predominant direction of flow. Ford et al. (1990) present a laterally averaged,

two-dimensional numerical model used to predict the vertical structure of the tidal current and

salinity profile in San Francisco Bay, which discretizes the vertical spatial scale into nine layers

through use of the σ-coordinate system. Smith (1987) compares one-dimensional numerical

model output to that produced by a 38-layer, laterally averaged, two-dimensional numerical

model, where both numerical models are applied to simulate the wind-driven flow patterns in a

coastal lagoon. Werner (1987) has developed a two-dimensional, finite element-based numerical

model which solves the viscous Navier-Stokes equations without the hydrostatic pressure

approximation in order to study wind-driven circulation over continental shelf edges as affected

by variations in bathymetry. Farrell and Stefan (1989) have modeled the inflow of a relatively

denser fluid into a reservoir fluid; Mendoza and Shen (1990) have simulated turbulent flow over

sand dunes with particular emphasis placed on the total flow resistance.

34

The need to use fully three-dimensional numerical models in order to better capture the

vertical structure of vertically stratified tidal flows is exemplified in the works of Sinha and

Sengupta (1987) and Jenter and Madsen (1989), where buoyancy-driven flow in rectangular

cavities is treated and an alternative bottom stress formulation is investigated, respectively.

Another such example work includes a study performed by Signell et al. (1990), who modeled

the effect of wind waves on wind-driven circulation. To this end, progress has been made

towards the three-dimensional modeling of tidal circulation, with most three-dimensional

numerical models making use of finite difference methods to solve the complete equations of

tidal motion; at present, finite element modeling in three dimensions remains a computationally

difficult task that must be constrained to relatively small domains or relatively coarse meshes

over relatively large domains.

The main distinguishing factor between two- and three-dimensional numerical models

involves the inclusion of property variations along the vertical spatial scale. However, due to the

small vertical grid lengths required to resolve the vertical spatial scale, a new class of

computational algorithms must be introduced in order to provide for suitable time steps of

computation. Improper treatment of the vertical spatial scale may lead to erroneous results

regarding vertical flow and chemical transport (Weaver and Sarachik, 1990). Meakin and Street

(1988a) provided suggestions for the treatment of complex domains through coordinate

transformations on an irregular region; Meakin and Street (1988b) then expanded on their overall

approach by splitting the complex domain into a number of geometrically simple overlapping

regions.

Leendertse (1989) presents an approach to three-dimensional, free-surface flow modeling

which includes appropriate approximation of the advective terms, finite difference solution

35

techniques, and a stability analysis. The governing equations presented neglect horizontal

momentum exchange; however, the computational algorithm leads to an efficient and stable

simulation in its test case application. Blumberg and Mellor (1987) present a numerical model

that sharply contrasts that of Leendertse (1989), which uses a vertical coordinate transformation,

turbulence closure, and a mode-splitting technique for its full implementation. Haidvogel et al.

(1990) apply a higher-order spectral technique over the vertical spatial scale in conjunction with

the σ-coordinate system and use a space-staggered, finite difference solution over horizontal

space in conjunction with an orthogonal curvilinear coordinate system. Bleck et al. (1989) have

developed an isopycnic-coordinate numerical model for ocean basin circulation, which is applied

to study mixed-layer thermocline interactions.

One region of the United States that is of specific hydraulic importance is Chesapeake

Bay, one of the largest estuaries in the world. Due to its particular significance, Chesapeake Bay

has been the subject of many modeling studies (Kim et al., 1990; Johnson et al., 1991). Both of

these studies employed a three-dimensional numerical model for curvilinear hydrodynamics

(CH3D) making use of boundary-fitted coordinates and turbulence closure. The CH3D

numerical model has also been extended to interface with an intertidal water-quality model for

Chesapeake Bay (Dortch et al., 1989). Other cases of three-dimensional, finite difference-based

modeling applications to estuarine systems include Gordon and Spaulding (1987), Isaji and

Spaulding (1987), Spaulding et al. (1987), Chu et al. (1989), and McCreary and Kundu (1989).

Initial applications of the finite element method to the simulation of surface water flow in

three dimensions led to the conclusion that this approach required excessive amounts of

computational expense when compared to the computer usage required to execute three-

dimensional, finite difference-based modeling applications (see Cheng et al. [1996], Cheng et al.

36

[1998a], and Cheng et al. [1998b]). However, in recent years, and primarily due to the

robustness of the GWCE formulation, three-dimensional, finite element-based numerical models

have demonstrated to be very useful tools for the modeling of tidal circulation. Lynch and

Werner (1987) present a linearized, harmonic numerical model, which is applied and verified for

a particular test case on Lake Maracaibo. In sequels to their work, Lynch and Werner (1991)

repeat the simulations using a non-linear, time-stepping numerical model, and Lynch et al.

(1990) compare two- and three-dimensional numerical model output to historical tidal data for

Lake Maracaibo. Their three-dimensional numerical model has also been applied to simulate the

tides occurring within the North Sea and English Channel (Lynch and Werner, 1988). Of

importance, while their results demonstrate the efficiency of the GWCE formulation in a variety

of three-dimensional modeling applications, the need for extensive velocity data sets to use

towards calibration and verification is emphasized.

3.2. Previous Modeling Studies for the Loxahatchee River Estuary

Although there exists an abundance of literature related to field studies involving the

Loxahatchee River estuary, most of these reports present analyses of salinity and flow

observational data and provide little, if any, insight towards developing a tidal model for the

estuary. (While the findings of these field studies are neglected in the following section of this

literature review, Chapter 5 contains historical information and empirical data from these reports

that are relevant to the present study.) To this end, three previous modeling studies involving the

Loxahatchee River estuary are covered for the purpose of reviewing past efforts that dealt with

37

studying the tides occurring within this estuarine system: Chiu (1975); Russell and Goodwin

(1987); Hu (2002).

Chiu (1975) conducted a saltwater intrusion study to determine the effect of removing

oyster bars that had recently formed in the vicinity of Jupiter Inlet. These oyster bars were

considered by local government and citizens to be a major cause of the deteriorating conditions

of the Loxahatchee River. It was supposed that the oyster bars were restricting tidal flow

through the inlet, which served to eliminate much of the self-cleaning capacity of the estuary.

Furthermore, the oyster bars were inhibiting boating by local residents and tourists.

After setting up a network of water level gages, current meters, and salinity monitoring

stations, Chiu (1975) set up and calibrated a numerical model to predict the effect on the

Loxahatchee River of removing the oyster bars that have been collecting around Jupiter Inlet.

The study concluded that dredging the affected areas of Jupiter Inlet to a depth of 2 m would

decrease the tidal range on the east side of Alternate A1A Bridge (see Figure 1.1) by about three

percent and delay the arrival of tidal flows by about 5 minutes; the tidal range on the west side of

Alternate A1A Bridge (see Figure 1.1) would increase by about three percent and the arrival of

tidal flows would advance by about 5 minutes. In addition, the numerical model predicted an

increase in peak flood tidal flow by about 9 cms in response to the clearing of Jupiter Inlet. This

increased inlet conveyance also served to move high-slack-water salinity profiles inland by about

100 to 250 m. Of importance, it is noted that the increased hydraulic conductivity created by

removing the oyster bars around Jupiter Inlet resulted in an enhanced tidal action within the

estuary and upstream movement of more saline waters into the Loxahatchee River.

Russell and Goodwin (1987) applied a two-dimensional, estuarine-simulation model

(SIMSYS-2D; see Leendertse [1970] and Leendertse and Gritton [1971]) to simulate tidal flows

38

and circulation patterns in the Loxahatchee River estuary. Their report presents results from one

objective of the overall study, which relate to the determination of the two-dimensional, tidally

induced circulation patterns occurring within the Loxahatchee River estuary. The information

gained from their modeling study served to help explain the distribution of bottom sediments and

waterborne constituents throughout the estuary.

The extent of the computational domain is described as extending from the nearshore

region of the Atlantic Ocean surrounding Jupiter Inlet, through the inlet entrance and central

embayment, up to the approximate upstream limit of tidal influence in the three forks of the

Loxahatchee River estuary. (Refer to Figure 1.1 for a map of the Loxahatchee River estuary,

highlighting these components of the estuarine system. Note that the extents of the map shown

in Figure 1.1 do not necessarily reflect the limits of the computational domain defined by Russell

and Goodwin [1987].) Parts of the AIW, both north and south of Jupiter Inlet, are also included

in the computational domain; however, the degree of coverage of the AIW is not clearly defined.

Russell and Goodwin (1987) commented that the north and south arms of the AIW were

initially modeled as water-storage areas; however, the evaluation of preliminary model results

indicated that while this assumption was adequate for the south arm of the AIW (because of low

tidal velocities), it was insufficient for the north arm of the AIW (due to higher tidal velocities).

A tidal boundary condition was then imposed on the north arm of the AIW, which served to

improve model results in subsequent simulations.

Tidal flows and circulation patterns computed for the Loxahatchee River estuary are

presented qualitatively as a series of vector maps (see Russell and Goodwin [1987]). Flood

transport patterns reveal large tidal flows through Jupiter Inlet towards the central embayment in

addition to significant transport rates flowing up the north arm of the AIW. The largest tidal

39

velocities are computed at the seaward side of the throat of Jupiter Inlet with the smallest tidal

velocities computed offshore and within the south arm of the AIW. (Ebb transport patterns are

analogous to those shown for flood tide, except that flow is directed seaward instead of

landward.) Further, vector maps of residual transport patterns in the vicinity of Jupiter Inlet

indicate net seaward tidal flows through the inlet channel and north arm of the AIW.

Russell and Goodwin (1987) conclude their study by remarking on the set up and

calibration of the SIMSYS-2D numerical model to simulate tidal flows and circulation patterns

in the Loxahatchee River estuary. Of importance, it is gathered from their modeling study that

the AIW plays a significant role in the distribution of tidal flows through the Loxahatchee River

estuary, namely within the coastal regions near Jupiter Inlet.

Hu (2002) presents field data analysis and discusses preliminary simulation output

obtained from a two-dimensional, hydrodynamic/salinity model for the Loxahatchee River

estuary (SFWMD, 2002). Major findings from Hu (2002) provide evidence to support the

premise that the advance of more saline waters up the Loxahatchee River is the result of the

combined effect of watershed hydrological changes, inlet modifications, and changes in MSL.

The amount of freshwater received by the Loxahatchee River estuary is a direct function of the

hydrological conditions of the Loxahatchee River watershed. (Refer to Chapter 5 for an

overview of past hydrological changes made to the Loxahatchee River watershed, which

highlights the highly transient nature of the freshwater river inflow conditions experienced in the

Loxahatchee River.) Further, Hu (2002) demonstrates that the quantity of freshwater river

inflow delivered to the Loxahatchee River estuary has a significant impact on the salinity regime

experienced within the Loxahatchee River, namely the upstream portions of the Northwest Fork.

40

As an aside, Hu (2002) also comments on the considerable difference in upstream salinity

levels as varying with the spring-neap tidal cycle (see Figure 2.1). Reporting these significant

responses in salinity to tidal fluctuations further supports the basis that such localized coastal

models require accurate tidal elevation boundary conditions in order to sufficiently capture the

physics of the processes being simulated.

The effect on the salinity regimes experienced within the Loxahatchee River as a result of

dredging material from Jupiter Inlet is then studied. Hu (2002) presents preliminary model

output which illustrates that deepening the inlet channel serves to push the salt wedge further

upstream the Northwest Fork. These simulated data also indicate that a shallower inlet reduces

the tidal influence within the Loxahatchee River.

The effects of MSL rise on the salinity regimes experienced within the Loxahatchee

River estuary are studied by performing a series of three simulations, varying MSL as follows:

current MSL; 100-years-earlier MSL; 100-years-later MSL. Holding all other run variables (e.g.,

freshwater river inflow input; inlet channel depth) constant for these three simulations, the

effects of MSL rise on salinity levels within the Loxahatchee River are isolated. Analysis of the

model results reveals that rising MSL serves to carry more saline waters further upstream the

Loxahatchee River, paralleling the salinity effects caused by deepening the inlet channel.

Similar to the findings of Chiu (1975), the increased hydraulic conductivity created by dredging

material from Jupiter Inlet (and rising MSL) results in an enhanced tidal action within the estuary

and upstream movement of more saline waters into the Loxahatchee River.

41

3.3. Tidal Asymmetry and Residual Circulation

A convenient example in which to introduce the concept of tidal asymmetry is through the

behavior of a shoreward approaching wind wave, which is characterized by a gradual steepening

of the wave front as the wave enters into shallower waters, followed by its sudden crash and

eventual run up along the shore. A similar distortion occurs for tidal waves approaching the

coast, however, the steepening of the wave fronts are not as observable (as those associated with

wind waves) due to the long periods associated with the tides. The essential requirement

necessary to produce this tidal distortion is that the wave amplitude be comparable to the depth

of water through which it is traveling.

Figure 3.1 displays an exaggerated profile of a wave being distorted as it moves into

shallower waters. At a fixed location, an observer will notice that it takes a longer time for the

water to fall than that required for the water to rise. The rate of rise of the water level is more

rapid than the rate of fall. This difference between the rates of water rise and fall increases as the

wave progresses (i.e., as x increases in Figure 3.1).

Figure 3.1. Distortion of a tidal wave propagating through shallow water up a channel in the

positive x-direction.

42

The tidal distortion shown in Figure 3.1 can be related to the propagation characteristics

of tidal waves traveling in the deep ocean. A tidal wave propagating through deeper waters (i.e.,

where the amplitude is significantly less than the depth D [so that shallow-water effects are not

contributing] and the depth D is small compared to the wavelength L [in practice, when

20LD < ; Open University, 2000]) travels at a speed:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1) gDc =

where g is the acceleration due to gravity. Since the wave speed decreases as the water depth

decreases, the troughs of the tidal wave will tend to be overtaken by the crests, which are

traveling through deeper water. This distortion of the tidal wave as it travels into shallower

waters gives rise to the tidal asymmetry that is illustrated in Figure 3.1.

Tidal currents flowing into and out of estuaries allow for a constantly changing regime of

sediment flux and coastal dynamics. The wave profile shown in Figure 3.1 is typical of that for a

tidal wave entering an estuary, where the wave steepening decreases the rise time and extends

the fall time. This tidal asymmetry acts to promote stronger flood tide currents and weaker ebb

tide currents. The sediment transported along the sea bed by currents (called the bed-load)

increases rapidly as the current speed increases, which means that in the case of Figure 3.1, more

sediment is carried inshore than is exported. More information regarding sediment transport and

net bed-loads in shallow estuaries, as caused by tidal asymmetries, can be found in Postma

(1967), Aubrey (1986), and Dronkers (1986b).

The study of tidal asymmetry in shallow estuaries and rivers has recently received a great

deal of attention (see LeBlond [1978], Boon and Byrne [1981], Parker [1984], Speer and Aubrey

43

[1985], Dronkers [1986a], and Friedrichs and Aubrey [1988]). The primary thrust of these

studies was based on examination of the mechanics of tidal propagation in shallow estuaries and

identification of the estuarine characteristics responsible for producing different types of tidal

asymmetries. A second goal of these investigations was formed by relating tidal asymmetry to

observed patterns of sediment transport and estuarine morphology. The result of this work

clarified the general causes of and mechanics involved in the generation of flood- and ebb-

dominant tidal asymmetries in shallow estuaries. (Flood dominance is characteristic of the

scenario depicted in Figure 3.1, where the duration of falling tides exceeds that of rising tides;

ebb dominance refers to the opposite situation.)

A distorted tide traveling up an estuary (e.g., the wave shown in Figure 3.1) can be

represented by the non-linear growth of higher harmonics and compounds of the principal

astronomical tidal constituents (see Table 2.2) (Dronkers, 1964; Uncles, 1981; Parker, 1984;

Aubrey and Speer, 1985). Even harmonics and compound tides formed from the principal

astronomical tidal constituents are capable of generating both time and magnitude asymmetries

in the tides observed within the estuary.

Along much of the Atlantic seaboard, the offshore tide is principally semi-diurnal in

character (see Figure 2.3), with the M2 tidal constituent acting in domination. When the M2

tidal constituent is the dominant semi-diurnal component, the M4 tidal constituent is the largest

quarter-diurnal tide formed within the estuary. Consequently, the ratio of the amplitudes (in both

sea surface elevation and velocity) of the M4 and M2 tidal constituents indicates the magnitude

of the tidal asymmetry generated within the estuary:

44

2

424

M

MMM H

HH = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.2)

where and are the amplitudes (in either sea surface elevation or velocity) of the M4

and M2 tidal constituents, respectively. Similarly, the relative phase of the M4 and M2 tidal

constituents determines the sense of tidal asymmetry (i.e., flood- or ebb-dominant):

4MH 2MH

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.3) 422422 MMMM ϕϕϕ −=−

where 22Mϕ and 4Mϕ are twice the phase of the M2 tidal constituent and the phase of the M4

tidal constituent, respectively. Relative phases (in sea surface elevation) between 0° and 180°

indicate a longer falling than rising tide, and hence, the tidal currents within the estuary tend to

be flood-dominant. Longer rising tides and ebb-dominant flow conditions are indicated by a

relative phase (in sea surface elevation) between 180° and 360°.

Table 3.1 lists amplitude ratios and relative phases (both in sea surface elevation) for the

five water level gaging stations located within the Loxahatchee River estuary (see Figure 1.1),

using the M4 and M2 tidal constituents extracted from the harmonic analysis presented in

Chapter 2. While the relative phases listed in Table 3.1 indicate that tidal flows through the

Loxahatchee River estuary should be flood-dominant (i.e., all computed values lie between

between 0° and 180°), the amplitude ratios shown in Table 3.1 reveal that the magnitude of this

flood dominance is very weak.

45

Table 3.1. Tidal asymmetry in the Loxahatchee River estuary, represented in terms of the M2-

M4 tidal constituent interaction.

Water level gaging stationa4MH (m)b

2MH (m)b24 MMH (-) 4Mϕ (°)b

2Mϕ (°)b422 MM −ϕ (°)

Coast Guard Dock 0.0032 0.3182 0.0101 320.78 8.70 56.62

Pompano Drive 0.0160 0.2986 0.0536 346.33 33.94 81.55

Boy Scout Dock 0.0194 0.3029 0.0640 354.97 43.11 91.25

Kitching Creek 0.0175 0.3077 0.0569 358.58 47.72 96.86

River Mile 9.1 0.0144 0.3037 0.0474 0.29 48.75 97.21 a Refer to Figure 1.1 for the locations of these five water level gaging stations. b Refer to Appendix D for a listing of the tidal constituent amplitudes and phases.

Two dimensionless parameters represent the principal estuarine characteristics

responsible for different types of tidal asymmetry. First, the ratio of the offshore amplitude of

the M2 tidal constituent a to the mean estuarine channel depth h measures the relative

shallowness of the estuary: ha . Second, the ratio of the volume of water stored between mean

high and low water in tidal flats and marshes Vs to the volume of water contained in channels at

MSL Vc measures the capacity of the estuary to store water as the tide rises from low to high

water (Friedrichs and Aubrey, 1988): cs VV .

Friedrichs and Aubrey (1988) suggest that tidal distortion in shallow estuaries results in a

compromise between two primary effects: frictional interaction of the tides with the channel

bottom; intertidal storage in tidal flats and marshes. The former effect is reflected in the ha

dimensionless parameter and leads to time delays between the ocean and estuary low water

exceeding the delays at high water (LeBlond, 1978; Dronkers, 1986a). The latter effect is

46

reflected in the cs VV dimensionless parameter and can be interpreted as a measure of the

efficiency of the exchange of water in the estuary around high water (Boon and Byrne, 1981).

The magnitude of the tidal asymmetry is controlled primarily by ha in flood-dominant

estuaries and cs VV in ebb-dominant estuaries. Table 3.2 lists computed values for the ha

dimensionless parameter for the five water level gaging stations located within the Loxahatchee

River estuary (see Figure 1.1), using the (amplitudes of the) M2 tidal constituents extracted from

the harmonic analysis presented in Chapter 2. These low values for ha further support the

weakness of the flood dominance of the Loxahatchee River estuary as determined from Table

3.1. (The cs VV dimensionless parameter is not computed for the Loxahatchee River estuary

due to the lack of topographic data, which would be used in calculating the storage capacity of

any tidal flats or marshes surrounding the estuarine system.)

Table 3.2. Magnitude of the tidal asymmetry in the Loxahatchee River estuary, represented in

terms of the ha dimensionless parameter.

Water level gaging stationa a (m)b h (m) ha (-)

Coast Guard Dock 0.3182 5.52 0.0576

Pompano Drive 0.2986 1.73 0.1726

Boy Scout Dock 0.3029 1.55 0.1954

Kitching Creek 0.3077 1.37 0.2246

River Mile 9.1 0.3037 1.37 0.2217 a Refer to Figure 1.1 for the locations of these five water level gaging stations. b Refer to Appendix D for a listing of the tidal constituent amplitudes and phases.

47

An alternative to using harmonic data for the evaluation of tidal asymmetry is to compute

residual circulation from numerical model output. Winant and Gutierrez de Velasco (2003)

define the average tidal cycle (ATC) as the average of any property as a function of tidal phase,

which is computed by dividing time-series data into sections of length equal to the period of the

M2 tidal constituent and averaging the sections. While Winant and Gutierrez de Velasco (2003)

use this ATC approach to examine the tidal asymmetries induced by the different terms involved

in bottom stress, residual circulation may be calculated using the ATC of globally computed

velocity vectors (see Russell and Goodwin [1987]). The resulting residual circulation patterns

would then be representative of the net tidal flows occurring within the estuary and provide

information relating to the flood or ebb dominance of the overall tidal circulation.

48

CHAPTER 4. NUMERICAL MODEL DOCUMENTATION

In modeling tidal flow and circulation within oceanic and coastal water bodies, set up of the

problem involves a thorough description of the physical system and phenomena being modeled

(e.g., spatial and temporal domain representations, approximations of the simulated processes,

characterizations of the applied boundary forcings) in a numerical setting. Due to the long-wave

nature of the sea surface response, as resulting from the tide-generating forces, the shallow-water

equations may be used to adequately describe the associated water level variations and

circulation patterns (see Leendertse [1967], Wang and Connor [1975], Lynch [1983], Spaulding

[1984], Smith and Cheng [1987], Walters [1987], Werner and Lynch [1987], Vincent and Le

Provost [1988], Signell [1989], Westerink et al. [1989], and Westerink and Gray [1991]). These

shallow-water equations describe mass and momentum conservation in a fluid and are valid

under the following assumptions: 1) the fluid must be vertically well-mixed with a hydrostatic

pressure gradient and constant density; 2) water waves of long wavelengths must be studied.

The former requirement holds for certain coastal regions and estuaries, and is an assumption to

be tested in the present study. The latter assumption eliminates the description of short-wave

phenomena where vertical acceleration is significant. Further, for tidal flows with horizontal

length scales that are large compared to the height of the vertical water column, the viscosity

terms may be assumed to be physically negligible (Dronkers, 1964; Blumberg and Mellor,

1987); however, in cases where residual circulation or tidal distortion within shallow-water

bodies is to be investigated, these non-linear advective terms cannot be ignored (Reid, 1990).

48

Tidal simulations are performed using ADCIRC-2DDI, the depth-integrated option of a

set of two- and three-dimensional fully non-linear hydrodynamic codes named ADCIRC

(Luettich et al., 1992). ADCIRC-2DDI uses the vertically integrated equations of mass and

momentum conservation, subject to incompressibility, Boussinesq, and hydrostatic pressure

approximations. For the applications presented in this study, the hybrid bottom friction

formulation is used, baroclinic terms are neglected, and lateral diffusion/disperson effects are

(when noted) employed, leading to the following set of balance laws in primitive, non-

conservative form, expressed in a spherical coordinate system (Flather, 1988; Kolar et al.,

1994a):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.1) ( ) 0coscos1

=⎥⎦

⎤⎢⎣

⎡∂

∂+

∂∂

+∂∂

φφ

λφζ VHUH

Rt

( ) UH

MH

gp

R

VfUR

UVR

UURt

U

SS∗−++⎥

⎦

⎤⎢⎣

⎡−+

∂∂

−

=⎟⎠⎞

⎜⎝⎛ +−

∂∂

+∂∂

+∂

τρτ

αηζρλφ

φφλφ

λλ

00

1cos1

tan1cos1∂

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.3) ( ) VH

MH

gp

R

UfUR

VVR

VURt

V

SS∗−++⎥

⎦

⎤⎢⎣

⎡−+

∂∂

−

=⎟⎠⎞

⎜⎝⎛ ++

∂∂

+∂∂

+∂

τρτ

αηζρφ

φφλφ

φφ

00

11

tan1cos1∂

where depth-integrated momentum dispersion in the longitudinal and latitudinal directions,

respectively, is given by (Blumberg and Mellor, 1987; Kolar and Gray, 1990):

49

( ) ( )⎥⎦

⎤⎢⎣

⎡∂

∂+

∂∂

= 2

2

2

2

22,,,

cos12

φλφφλHVUHVU

RE

M h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.4)

and t = time; λ, φ = degrees longitude (east of Greenwich positive) and latitude (north of equator

positive), respectively; U, V = depth-integrated velocity in the longitudinal and latitudinal

directions, respectively; H = total height of the vertical water column, h + ζ; h = bathymetric

depth, relative to MSL; ζ = free surface elevation, relative to MSL; R = radius of the Earth;

=Ω= φsin2f Coriolis parameter; Ω = angular speed of the Earth; pS = atmospheric pressure at

the free surface; ρ0 = reference density of water; g = acceleration due to gravity; α = effective

Earth elasticity factor; = horizontal eddy viscosity; 2h

E λτ S , φτ S = applied free surface stress in

the longitudinal and latitudinal directions, respectively; ∗τ = quadratic bottom stress; η =

Newtonian equilibrium tide potential.

A formal development of the tidal potential (after Doodson [1921], Cartwright and

Taylor [1971], and Cartwright and Edden [1973]) is provided in Appendix A; however, a

practical expression for the Newtonian equilibrium tide potential is also given by Reid (1990):

( ) ( ) ( ) ( )∑⎥⎥⎦

⎤

⎢⎢⎣

⎡++

−=

jnjn

jnjjnjnjn uj

TttLtfHt

,

00

2cos,, λπφαφλη . . . . . . . . . . . . . . . . . . . . . . . . . . (4.5)

where the latitude-dependent functions, ( )φjL , for the tidal species j (0, 1, 2 = long-period,

diurnal, semi-diurnal) are given by:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.6) 1sin3 20 −= φL

50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.7) φ2sin1 =L

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.8) φ22 cos=L

and t0 = reference time; Hjn, Tjn = equilibrium amplitude and period of tidal constituent n of

species j, respectively (see Table 2.2); fjn and ujn = time-dependent nodal factor and equilibrium

argument, respectively (see Schureman [1941] and Schwiderski [1980]). The gradient of αη

results in the effective tide-producing force (see Appendix A). The effective Earth elasticity

factor α accounts for the reduction in the field of gravity due to the existence of small tidal

deformations of the Earth’s surface (called Earth tides). The value α = 0.69 is the ratio of the

theoretical period of the Earth’s wobble derived by Euler (assuming the Earth to be a perfectly

rigid sphere) to the observed period of the Earth’s wobble (Reid, 1990). (Therefore, α is a

measure of the rigidity of the Earth, and for reference, α = 1 would correspond to a perfectly

rigid sphere.) In modeling global ocean tides, Schwiderski (1980) and Hendershott (1981)

recommend α = 0.69, although the value of the effective Earth elasticity factor has been shown to

be slightly dependent upon the tidal constituent (Wahr, 1981).

To facilitate finite element-based solutions to Eqs. (4.1)-(4.3), ADCIRC-2DDI maps the

governing equations from spherical form into a rectilinear coordinate system using a Carte

Parallelogrammatique (CP) projection (Pearson, 1990):

( ) 00 cosφλλ −=′ Rx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.9)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.10) φRy =′

51

where ( )00 ,φλ is the center of the projection. Applying the CP projection to Eqs. (4.1)-(4.3) and

neglecting lateral diffusion/dispersion effects gives the shallow-water equations in primitive,

non-conservative form, expressed in the CP coordinate system:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.11) ( ) 0coscos

1coscos 0 =

′∂∂

+′∂

∂+

∂∂

yVH

xUH

tφ

φφφζ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.12) ( ) UH

gp

x

VfURy

UVxUU

tU

SS∗−+⎥

⎦

⎤⎢⎣

⎡−+

′∂∂

−

=⎟⎠⎞

⎜⎝⎛ +−

′∂∂

+′∂

∂+

∂

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.13)

Utilizing the finite element method to resolve the spatial dependence of the shallow-water

equations in their primitive form gives inaccurate solutions with severe artificial (near 2Δx)

modes (Gray, 1982). Therefore, the primitive balance laws are rewritten into the GWCE to

provide highly accurate, noise free, finite element-based solutions the shallow-water equations

(Lynch and Gray, 1979; Platzman, 1981; Foreman, 1983; Kinnmark, 1985; Gray, 1989; Walters

and Werner, 1989; Werner and Lynch, 1989). The GWCE is derived by combining a time-

differentiated form of the primitive continuity equation and a spatially differentiated form of the

primitive, momentum equations (recast into conservative form), and adding to this result, the

primitive continuity equation multiplied by a constant in time and space, τ0, followed by a

∂

τρτ

ηζρφ

φ

φφφ

λ

00

0

0

coscos

tancoscos

( ) VH

gpy

UfURy

VVxVU

tV

SS∗−+⎥

⎦

⎤⎢⎣

⎡−+

′∂∂

−

=⎟⎠⎞

⎜⎝⎛ ++

′∂∂

+′∂

∂+

∂∂

τρτ

ηζρ

φφφ

φ

00

0 tancoscos

52

transformation of the advective terms into non-conservative form (Lynch and Gray, 1979;

Kinnmark, 1985; Luettich et al., 1992; Kolar et al., 1994b). The GWCE is expressed in a

spherical coordinate system as:

. . . . . . . . . . . . . . . . . . . . . . . . . (4.14)

( ) ( )

( )

( ) 0tan

tancoscos11

coscos

tancoscos1

cos1

000

2

0

00

2

0

02

2

2

2

=⎟⎠⎞

⎜⎝⎛ +

∂∂

−⎭⎬⎫

−−+∂∂

∂+

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−+

∂∂

−⎟⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡∂

∂+

∂∂

∂∂

and in the CP coordinate system (neglecting lateral diffusion/dispersion effects) as:

. . . . . . . . . . . . . . . . . . . . . . . . . . (4.15)

−

⎭⎬⎫

−−+∂∂

∂+⎥

⎦

⎤⎢⎣

⎡−+

∂∂

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

∂∂

−∂∂

+∂

∗

∗

VHtVH

RVH

tRE

gpRHUHfU

RHVVHUV

RR

UHtR

Egp

RH

VHfUR

UVHUUHRRtt

Sh

S

ShS

τφττρτ

φζ

αηζρφ

φφ

φλφφ

ττρτ

λζ

φαηζ

ρλφ

φφ

φλφλφ

ζτζ

φ

λ

∂

−

( ) ( )

( )

( ) 0tantan

tancoscos

coscos

tancoscos

coscos

00

0

0

0

00

0

0

0002

2

=⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

∂∂

−⎭⎬⎫

+−−

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−+

′∂∂

−⎟⎠⎞

⎜⎝⎛ +−

′∂∂

−′∂

∂−

∂∂

′∂∂

+

⎭⎬⎫

+−−⎥⎦

⎤⎢⎣

⎡−+

′∂∂

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ ++

′∂∂

−′∂

∂−

∂∂

′∂∂

−∂∂

+∂

∗

∗

VHR

VHRt

VH

gp

yHUHfU

RyVVH

xVUH

tV

y

UHgp

xH

VHfURy

UVHxUUH

tU

xtt

S

S

SS

φτφρτ

ττ

ηζρ

φφφζ

ρτ

ττηζρφ

φ

φφφζ

φφζτζ

φ

λ

∂

−

53

where τ0 = GWCE weighting parameter (i.e., for large values of τ0, the GWCE reverts to the

primitive continuity equation; for small values of τ0, the GWCE acts as a pure wave equation).

The GWCE is solved in conjunction with the primitive, non-conservative momentum equations.

The high accuracy of this formulation (using the GWCE) is a result of its excellent

numerical amplitude and phase propagation characteristics. In fact, Fourier analysis indicates

that in waters of constant bathymetric depth and using linear interpolation, a linear tidal wave

with 25 nodes per wavelength is more than adequately resolved over the range of Courant

numbers (Westerink et al., 1994a):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.16) ( ) 0.1# ≤ΔΔ= xtghC

where Δt and Δx correspond to the applied time step and nodal spacing, respectively; h relates to

the bathymetric depth; g is the acceleration due to gravity. Furthermore, the monotonic

dispersion behavior of this approach (using the GWCE) avoids generating artificial (near 2Δx)

modes which plague primitive-based finite element solutions. It is noted that the monotonic

dispersion behavior of GWCE-based finite element solutions is very similar to that associated

with staggered finite difference solutions to the primitive shallow-water equations (Westerink

and Gray, 1991). GWCE-based finite element solutions to the shallow-water equations allow for

extremely flexible spatial discretizations, which results in a highly effective minimization of the

discrete size of the problem (Le Provost and Vincent, 1986; Foreman, 1988; Vincent and Le

Provost, 1988; Westerink et al., 1992a).

The numerical discretization of the GWCE and non-conservative momentum equations

has been implemented using strategies similar to Werner and Lynch (1987) and Kolar and Gray

54

(1990) and is described in detail by Luettich et al. (1992), Kolar et al. (1994a), and Kolar et al.

(1994b). The discretization procedure is implemented in three well-defined stages. First,

symmetrical weak weighted residual statements are developed for the GWCE and non-

conservative momentum equations. The resulting equations require C0 functional continuity.

Second, the equations are time discretized. A variably weighted three-time-level implicit

scheme is used for most linear terms in the GWCE with the non-linear, Coriolis, atmospheric

pressure forcing, and tidal potential terms treated explicitly. The time derivative term that

appears in the non-conservative advective terms in the GWCE is evaluated at two known time

levels. A Crank-Nicolson two-time-level implicit discretization is applied to all of the terms in

the non-conservative momentum equations with the exception of the bottom stress, advective,

and eddy viscosity terms, which are treated explicitly.

Finally, the finite element method is implemented, which involves the following: the

variables (free surface elevation, depth-integrated velocity, bathymetric depth) are expanded over

C0 three-node linear triangles (with the exception of the non-spatially differentiated portion of

the advective terms in the final weighted residual form of both the GWCE and non-conservative

momentum equations, which apply L2 interpolating functions); discrete equations on an

elemental level are developed; global systems of equations are assembled.

Depth forcings are applied in the discrete GWCE and normal-flux boundary conditions

are enforced in the discrete, non-conservative momentum equations. Westerink et al. (1994d)

have shown that solutions to the GWCE are insensitive to this standard boundary condition

formulation. It should also be noted that the discrete GWCE is decoupled from the discrete, non-

conservative momentum equations, allowing for a sequential solution procedure to follow.

Furthermore, the GWCE system matrix is independent of time and only requires assemblage and

55

decomposition once for a direct solver. Mass lumping is implemented for the non-conservative

momentum equations. Therefore, even though the system matrix for the discrete, non-

conservative momentum equations is dependent upon time, it is trivial to solve since it is

diagonal. These features that have been described make ADCIRC-2DDI highly efficient in

terms of computational requirements.

The manner in which bottom friction is parameterized significantly affects the

contribution of bottom stress to the overall propagation of the tides. In general, most two-

dimensional numerical models use either a standard quadratic or a Manning’s type bottom

friction formulation, both of which are functions of the depth-integrated velocity:

HVUC f

22 +=∗τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.17)

where Cf = bottom friction factor. In applying a Manning’s type bottom friction formulation, the

bottom friction factor may be computed using one of the following relationships (Luettich et al.,

1992):

8DW

ffC = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.18)

2C

f CgC = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.19)

31

2

hgnC M

f = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.20)

56

where fDW = Darcy-Weisbach friction factor; CC = Chezy friction coefficient; nM = Manning’s

friction factor.

Particular emphasis has been placed on understanding the influence of the quadratic

bottom friction parameterization in rivers (Godin, 1991a; Parker, 1991) and shallow seas

(Pingree, 1983; Speer and Aubrey, 1985; Pingree and Griffiths, 1987). More recently, studies

particularly focused on describing the specific effects of quadratic bottom friction within coastal

regions and estuaries (Godin and Martinez, 1994; Godin, 1999) have provided further insight

into the parameterization of bottom friction.

Despite considerable progress in shallow water modeling, numerous investigations

(Sidjabat, 1970; Snyder et al., 1979; Westerink et al., 1989) involving the application of two-

dimensional numerical models to coastal seas have shown inadequacies in using a quadratic

formulation to represent bottom friction. Results presented by Grenier et al. (1995) and Cobb

and Blain (1999) suggest the need for a frictional closure more advanced than the standard

quadratic bottom friction formulation in order to reduce the non-linear frictional effect relative to

the linear frictional effect, a damping problem commonly encountered in modeling shallow-

water systems.

Luettich et al. (1992) recommend the use of a hybrid formulation of the standard

quadratic bottom friction parameterization for hydrodynamic studies involving shallow-water

systems, which allows for the bottom friction factor to change with respect to bathymetric depth.

In very shallow waters, the hybrid bottom friction formulation is useful particularly when the

wetting and drying of elements is implemented since this expression becomes highly dissipative

as the water depth becomes small (Luettich et al., 1992). Murray (2003) demonstrates the

advantages of using this hybrid bottom friction formulation in a study where the wetting and

57

drying of elements is employed. Hagen et al. (2005a) expand on this study by examining the

flow dependence of the minimum bottom friction factor as used in the hybrid bottom friction

formulation:

θγ

θ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=HHCC break

ff 1min

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.21)

where = minimum bottom friction factor that is approached in deep waters when the hybrid

bottom friction formulation reverts to a standard quadratic bottom friction function; H

minfC

break =

break depth to determine if the hybrid bottom friction formulation will behave as a standard

quadratic bottom friction function or increase with water depth similar to a Manning’s type

bottom friction function; θ = dimensionless parameter that establishes how rapidly the bottom

friction factor approaches its upper and lower limits; γ = dimensionless parameter that describes

how quickly the bottom friction factor increases as water depth decreases.

Luettich et al. (1992) recommend values of 10 m, 10, and 1/3 for Hbreak, θ, and γ,

respectively. Figure 4.1 displays the progression of the bottom friction factor from a larger value

(as governed by a Manning’s type bottom friction formulation) in shallower waters to the

minimum value (according to the specification of the minimum bottom friction factor) in deeper

waters for fixed values of the break depth and the two dimensionless parameters.

58

Figure 4.1. Depth-dependence of the hybrid bottom friction factor; see Eq. (4.21).

59

CHAPTER 5. PRESENTATION OF STUDY AREA

The Loxahatchee River estuary and its watershed are located along the southeastern coast of

Florida within the Lower East Coast Planning Area (SFWMD, 2000). The Loxahatchee River

watershed consists of approximately 550 km2 of natural, urban, suburban, and agricultural lands

and is located within northern Palm Beach and southern Martin counties. The central

embayment of the Loxahatchee River estuary is at the confluence of three major tributaries (see

Figure 1.1): the Northwest Fork (Loxahatchee River); the North Fork; the Southwest Fork. The

Loxahatchee River originates at the G-92 structure in northern Palm Beach county, flows north

to enter Martin county, continues north and bends east through Jonathan Dickinson State Park

(JDSP), and then forms a confluence with the North Fork and Southwest Fork at the central

embayment to connect to the Atlantic Ocean via Jupiter Inlet. The Atlantic Coastal Ridge (ACR)

in eastern Martin County defines the headwaters of the North Fork, which flows south-southeast

into the central embayment. All but 1.5 km of the Southwest Fork has been channelized to form

the C-18 canal, which flows northeast through Palm Beach county to discharge into the central

embayment.

The Loxahatchee River, which is often referred to as being the last free flowing river in

southeast Florida, and its upstream floodplains are unique regional resources in several ways.

On May 17, 1985, a 12-kilometer-long reach of the Loxahatchee River was federally designated

as Florida’s first Wild and Scenic River (Florida Department of Natural Resources, 1985). In

addition, different portions of the Loxahatchee River estuary were designated as an aquatic

preserve, Outstanding Florida Waters, and a state park. The Loxahatchee River represents one of

60

the last vestiges of native cypress river swamp within southeast Florida. Large sections of the

Loxahatchee River and its watershed are included within JDSP (see Figure 5.1[a]), which

contain outstanding examples of the region’s natural habitats.

The Loxahatchee River watershed is unique in that it contains a number of natural areas

that are essentially intact and in public ownership. These areas include the J.W. Corbett Wildlife

Management Area (CWMA), JDSP, Loxahatchee and Hungryland Sloughs (see Figure 5.1[a]),

Hobe Sound National Wildlife Refuge, Juno Hills and Jupiter Ridge Natural Areas, Pal-Mar, and

ACR. These natural areas contain pinelands, sand pine and xeric oak scrubs, hardwood

hammock, freshwater marsh, wet prairie, cypress and mangrove swamps, ponds, sloughs, rivers

and streams, seagrass and oyster beds, and coastal dunes, which support diverse biological

communities, including many protected species (FDEP, 1998).

Preservation and enhancement of these outstanding natural and cultural values form the

primary goals of the SFWMD’s management program for Loxahatchee River estuary. The

vision of the SFWMD for protecting the water resources of the Loxahatchee River estuary

include: 1) maintaining surface water and groundwater inflows to the Loxahatchee River; 2)

providing minimum freshwater river inflows to control upstream movement of the salt wedge

during dry season conditions; 3) preserving existing water quality in the Loxahatchee River by

eliminating identified water-quality problems; 4) supporting river discharges needed to sustain

natural systems within the downstream portions of the Loxahatchee River estuary. In addition,

the SFWMD and FDEP have jointly developed restoration proposals and are working with other

agencies, local interests, and concerned citizens to arrive at a practical plan for preservation and

enhancement of the Loxahatchee River estuary.

61

Figure 5.1. Map of the Loxahatchee River watershed (after FDEP [1998]) highlighting (a) the

boundaries of JDSP and the Loxahatchee and Hungryland Sloughs and the layout of

the local road/highway system along with (b) the margins of the seven major

drainage sub-basins located within its interior.

62

The southeastern coastal region of Florida experiences a subtropical climate with daily

temperatures ranging from an average of 28°C in the summer to an average of 19°C in the

winter; the average annual temperature is around 24°C (Breedlove Associates, Inc., 1982).

Prevailing winds from the east and southeast provide a marine influence, with average wind

speeds of approximately 16 km/hr. The air masses over the region are generally moist and

unstable, which leads to frequent rain showers, usually of short duration, with summertime

thundershowers occurring, on average, every other day. Rainfall over the Loxahatchee River

watershed averages about 155 cm annually with a median annual rainfall rate of about 145 cm

(Breedlove Associates, Inc., 1982). Dent (1997) reports that since the early 1960s, about two-

thirds of this precipitation occurs during the wet season (May through October), while the

remaining one-third of this precipitation falls during the dry season (November through April).

On average, maximum monthly rainfall amounts of 22 cm occur during the month of September,

while minimum monthly rainfall amounts of around 6 cm occur during the months of December,

January, and February (SFWMD, 1998). May and November are transitional months and

sometimes represent key time periods for either prolonging or relieving drought/flood conditions

(Dent, 1997). Dent (1997) also provides information about the spatial distribution of

precipitation over the Loxahatchee River watershed, which indicates that wet season rainfall is

higher inland as compared to the amount of precipitation received nearer the coast. These

findings are similar to those of MacVicar (1981) whom reported that the predominance of

convective type rainfall in South Florida during the wet season results in much higher rainfall

amounts on the mainland than near the ocean shore.

The Loxahatchee River historically received freshwater river inflow at the upstream end

of the Northwest Fork from the Loxahatchee and Hungryland Sloughs (Parker et al., 1955). Both

63

of these wetland areas drained to the north from the low divides near State Road 710 (see Figure

5.1[a]). Historically, this area was characterized by swampy flatlands interspersed with small,

often interconnected, ponds and streams that produced sheet flow that might be directed north or

south, depending on local conditions. Drainage patterns were determined by the poorly defined

natural landforms of the area.

The major features that presently influence drainage in the Loxahatchee River watershed

are the C-18 canal (see Figure 1.1) and the Florida Turnpike, Interstate 95, and State Roads 710

and 708, which act as important sub-basin divides (see Figure 5.1[a]), and the extensive system

of secondary canals developed by special drainage districts and landowners within the river

basin. Since the turn of the century, human activities have altered nearly all of the natural

drainage patterns within the river basin. Many areas that once were wetlands, ponds, and

sloughs are now a network of drainage canals, ditches, roads, highways, well-drained farms,

citrus groves, golf courses, and residential developments. This drainage network has

significantly altered surface water inflows to the Loxahatchee River estuary and lowered

groundwater levels within the surrounding watershed (McPherson and Sabanskas, 1980). During

the years of 1957 and 1958, the C-18 canal was constructed through the central portion of the

Loxahatchee Slough (the former headwaters of the Loxahatchee River) for flood protection

purposes. This project redirected freshwater river inflows from the Northwest Fork to the

Southwest Fork from the early 1960s up to 1974, when the G-92 structure was constructed to

reconnect the C-18 canal and Loxahatchee Slough with the Northwest Fork (see Figure 1.1).

The Loxahatchee River historically drained 700 km2 of inland sloughs and wetlands.

Some of the major tributary systems (e.g., the North Fork, the Northwest Fork, and Kitching

Creek) exist today largely within their historical river banks. Other creeks (e.g., the Southwest

64

Fork, Cypress Creek, and Hobe Grove Ditch) have been greatly altered. Today, the Loxahatchee

River watershed encompasses about 80 percent of its historical size (about 550 km2 in areal

coverage). More than half of the land still remains undeveloped and the remainder has been

altered by agricultural or urban development. Undeveloped lands consist of wetlands and

uplands. The Loxahatchee River watershed also contains about 16 km2 of open water including

lakes and the estuary (FDEP, 1998).

Although the total area of the Loxahatchee River watershed has not changed dramatically,

drainage patterns have been significantly altered due to road construction (e.g., State Road 710,

Interstate 95, Florida Turnpike), construction of the C-18 canal and associated water-control

structures, and the development of an extensive canal network. The canal network was designed

primarily to provide drainage and flood protection for agricultural and urban development and

associated water conveyance for potable use and irrigation; however, these modifications made

to the Loxahatchee River watershed have altered natural flow regimes and drainage patterns and

lowered groundwater levels throughout the river basin.

The Loxahatchee River watershed consists of seven major drainage sub-basins, which

provide surface water inflows to the three forks of the Loxahatchee River estuary. The sub-basin

boundaries are based primarily on hydrology and secondarily on land use (see Figure 5.1[b]).

Each of these seven sub-basins plays an important role in the drainage processes of the

Loxahatchee River watershed. Table 5.1 lists and provides descriptions of the seven major

drainage sub-basins found within the Loxahatchee River watershed.

65

Table 5.1. 7 major drainage sub-basins of the Loxahatchee River watershed (after FDEP

[1998]).

Sub-basin Size (km2) Land use and drainage characteristics

JDSP/Hobe Sound

93

Runoff from natural lands within this sub-basin is partially discharged into the North Fork, with the remaining surface water inflow supplied to Kitching Creek (which then flows into the Northwest Fork).

Coastal 88 Runoff from highly developed lands drains into the AIW (which is then carried into the Atlantic Ocean through Jupiter Inlet) providing discharges to the marine waters of the Loxahatchee River estuary rather than to the freshwater portions of the Northwest Fork.

Estuary 54 A significant amount of runoff contributed to the brackish waters of the central embayment make this sub-basin the central drainage system of the Loxahatchee River estuary.

C-18 Canal/ J.W. CWMA

259 Includes remnants of the Loxahatchee and Hungryland Sloughs, which historically fed the Northwest Fork. Today, surface water inflows are carried by the C-18 canal and discharged either into the Southwest Fork or through the G-92 structure into the upstream end of the Northwest Fork.

Cypress Creek/ Pal-Mar

119 Drains a sizeable wetland area located in the western extremities of the Loxahatchee River watershed to provide surface water inflows to Cypress Creek (which then flow into the Northwest Fork).

Groves 44 Altered to support agricultural (mostly citrus) production to provide a valuable greenway link between natural areas located within the Loxahatchee River watershed. Surface water inflows are discharged into Hobe Grove Ditch (which then flow into the Northwest Fork).

Wild and Scenic River/ Jupiter Farms

60 Substantial rural development (Jupiter Farms) characterizes the upstream section of this sub-basin; the downstream section of this sub-basin comprises the protected reach of the Loxahatchee River. Runoff from this sub-basin is discharged into the upstream portions of the Northwest Fork.

The Loxahatchee River estuary is divided into three components in order to establish

minimum flows and levels for the Loxahatchee River: 1) the Northwest Fork (namely its

protected reach) and its upstream floodplains, which include the Loxahatchee Slough, JDSP,

66

Cypress Creek, Hobe Grove Ditch, and Kitching Creek; 2) downstream areas of the Loxahatchee

River estuary, including the central embayment, the North Fork, and the Southwest Fork; 3)

coastal waters of the AIW and within Jupiter Inlet. The Northwest Fork originates in the

Loxahatchee Slough, which receives discharges from the C-18 canal and runoff and groundwater

inflows from adjacent uplands. Downstream from the Loxahatchee Slough, the Northwest Fork

receives additional surface water inflows from three major tributaries (see Figure 1.1): 1)

Cypress Creek, which drains a portion of the Cypress Creek/Pal-Mar sub-basin (see Table 5.1);

2) Hobe Grove Ditch, which drains a portion of the Groves sub-basin (see Table 5.1); 3)

Kitching Creek which drains wetlands found north of the Loxahatchee River (see Table 5.1).

The Northwest Fork flows through cypress swamp, mangrove forest, and JDSP to the saline

waters of the lower portions of the Loxahatchee River estuary. Much of the surrounding

watershed remains in a natural state or for low-intensity agricultural use so that the water quality

of runoff from most areas is good. Large tracts of the surrounding watershed are protected in

parks or preserves, and additional land is being purchased by various private interests and

government entities for preservation purposes.

The floodplain of the Northwest Fork is a prime example of a pristine subtropical river

cypress swamp and represents a last vestige of this community within southeast Florida (U.S.

Department of the Interior and National Park Service, 1982). The cypress swamp community

extends 6.5 km (downstream from State Road 706) along the Northwest Fork. Originally, the

cypress forest extended further downstream to a point beyond the confluence with Kitching

Creek. Today, as a result of saltwater intrusion up the Northwest Fork, freshwater cypress and

hardwood communities share the adjacent floodplains with saltwater-tolerant mangroves. The

remaining cypress swamp community exhibits high species diversity due to the overlap of

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tropical and temperate zone communities. Tropical vegetation (e.g., wild coffee, myrsine,

leather fern, and cocoplum) may be found along with pop ash, water hickory, red bay, royal fern,

and buttonbush, which are considered to be more northern flora (U.S. Department of the Interior

and National Park Service, 1982). The slightly elevated areas that border the Northwest Fork are

dominated by slash pine and saw palmetto, in addition to some scrub oaks and many herbs and

grasses. Threats to floodplain vegetation include periods of saltwater intrusion within upstream

areas of the Loxahatchee River, which result in death or stress to the remaining freshwater

species and replacement by saltwater-tolerant species (e.g., red mangroves, Brazilian pepper, and

climbing ferns).

The spatial distribution of major vegetation communities found along the Northwest Fork

during the early 1940s, 1985, and 1995 has been documented in SFWMD (2002) to better

understand the response of these major vegetation communities to land use changes as have

occurred over the past half-century. The following summary provides an overview of this

documentation. Aerial photographs taken in the early 1940s revealed an abundant presence of

swamps, wet prairies, and inland ponds and sloughs. Mangroves (representing 23 percent of the

vegetative coverage of the Northwest Fork) dominated the downstream watershed areas along

the Northwest Fork while freshwater cypress swamp communities (comprising 73 percent of the

vegetative coverage of the Northwest Fork) were present further upstream of these saltwater-

tolerant species. An apparent reduction in total coverage of the river floodplains between the

early 1940s and 1995 can be attributed to several causes, including the scouring of the riverbed,

bulkheading, development, and loss of wetland vegetation to transitional and upland species (as

due to saltwater intrusion up the Loxahatchee River). By 1985, much of the watershed had been

developed with the exception of JDSP. Freshwater vegetation represented 61 percent of the total

68

area with mangroves accounting for 25 percent of the vegetative coverage. Mangroves

experienced only a 4 percent increase in overall vegetative coverage due to floodplain

urbanization while freshwater cypress swamp communities decreased in overall vegetative

coverage by 10 percent. Freshwater river inflows delivered to the Northwest Fork increased

during the period between 1985 and 1995 due to the construction and improved operation of the

G-92 structure and larger rainfall amounts. (These watershed changes may account for the fact

that only minor differences in vegetation patterns occurred during this ten-year period.)

Improved aerial photography that was used in 1985 and 1995 made it possible to distinguish

differences in structure and composition of the cypress swamp communities, which further

indicated the adverse effects of saltwater intrusion on this freshwater vegetation.

Upon the designation of the (upper 12 km of the) Northwest Fork as a Wild and Scenic

River, special considerations were taken to ensure that the surrounding watershed remained

protected by maintaining sufficient inflow conditions, good water quality, and natural floodplain

areas. A number of goals were identified for the Loxahatchee River watershed to address these

protection issues (FDEP and SFWMD, 2000). Of particular importance, the development of

river-discharge criteria to preserve the historical freshwater communities within the Loxahatchee

River was initiated (SFWMD, 2002). Major sources of freshwater river inflow to the

Loxahatchee River include Lainhart Dam (through the G-92 structure), Cypress Creek, Hobe

Grove Ditch, and Kitching Creek (see Figure 1.1). Of these four tributaries, Lainhart Dam (in

operation with the G-92 structure) provides surface water inflows to the main stem of the

Loxahatchee River and is the largest contributor (supplying between 51 and 56 percent of the

total freshwater river inflow received by the Northwest Fork) of these surface water inflows.

The second largest contributor (supplying between 26 and 32 percent of the total freshwater river

69

inflow received by the Northwest Fork) of surface water inflow to the Loxahatchee River is

Cypress Creek, followed by Kitching Creek (between 11 and 13 percent) and Hobe Grove Ditch

(about 5 percent). In terms of water-control management, the G-92 structure represents not only

the largest source of surface water inflows delivered to the Northwest Fork, but also the only

water-control structure that can be operated by the SFWMD to increase or decrease freshwater

river inflow to the Loxahatchee River. Surface water inflows supplied by Kitching Creek are

currently unregulated and are largely driven by rainfall. Cypress Creek and Hobe Grove Ditch

contain water-control structures that are operated by other drainage districts.

In the early 1900s, Jupiter Inlet was artificially opened on several occasions. In 1921, the

Jupiter Inlet District (JID) was established and provided oversight for dredging of Jupiter Inlet in

1922, 1931, 1936, and every few years after 1947. Dredge and fill operations have also been

carried out in the central embayment and within the three adjoining forks. Further information

regarding past dreading activities within the Loxahatchee River estuary is provided by

McPherson et al. (1982). McPherson et al. (1982) also discuss the influence of sedimentation

and erosion processes on the bathymetry of the Loxahatchee River estuary, noting the

development of a large horseshoe-shaped sand bar within the central embayment over the

twenty-year period from 1960 to 1980 as a prime example of the bathymetric alterations caused

by sediment transport and deposition.

The United States Geological Survey (USGS) measured the incoming and outgoing tidal

volumes within the Loxahatchee River estuary for several days in 1980. It was determined that,

on average, 57 percent of the incoming tidal volume at Jupiter Inlet flowed into the Loxahatchee

River estuary west of the Alternate A1A Bridge. (See Figure 1.1 for the location of the Alternate

A1A Bridge as it transects over the central embayment.) McPherson et al. (1982) calculated a

70

mean tidal prism† of 4 million m3 for the Loxahatchee River estuary using data measured at the

Alternate A1A Bridge. This tidal volume accounts for about 63 percent of the total volume of

water contained within the Loxahatchee River estuary (west of the Alternate A1A Bridge). In a

related study, Chiu (1975) reported that 45 percent of the total tidal exchange entered the

Loxahatchee River estuary, while the remaining portion entered the north and south arms of the

AIW. These findings indicate that freshwater river inflow provided to the Loxahatchee River

estuary is very small compared to the exchange of tidal volumes. McPherson et al. (1982)

reported that dry and wet season freshwater river inflows (as supplied by the three forks of the

Loxahatchee River estuary) represent about 1 and 5 percent of the tidal prism (see footnote on

current page) (as corresponding to the data measured at the Alternative A1A Bridge),

respectively. Of the total freshwater river inflow volume, 77 percent is discharged into the

Northwest Fork, 21 percent is carried by the Southwest Fork, and the remaining 2 percent flows

through the North Fork (McPherson et al., 1982).

The central embayment is shallow with average and maximum depths of 1.0 and 4.5 m,

respectively, covering an area of approximately 1.5 km2 (Russell and McPherson, 1984;

Antonini et al., 1998; FDEP, 1998). The waters of the central embayment are tidally dominated,

receiving, on average, only 8 and 5 cms of freshwater river inflow from all upstream sources

during wet and dry season conditions, respectively. Analysis of historical patterns of seagrass

and oyster reef populations within the central embayment suggests that this section of the

Loxahatchee River estuary has experienced highly variable salinity regimes, which may mostly

be attributed to the periodic opening and closing of Jupiter Inlet (Antonini et al., 1998).

† U.S. Army Corps of Engineers (2002) defines the tidal prism as the volume of water that enters through an inlet channel during flood flow or exits through an inlet channel during ebb flow (whichever is greater).

71

The central embayment serves as a confluence for the three major tributaries of the

Loxahatchee River estuary: the Northwest Fork; the North Fork; the Southwest Fork. The

Northwest Fork has been considerably altered from its original condition due to development

along the coastline and dredging activities. The Northwest Fork is a natural river channel with

depths generally ranging from 1 to 2 m deep (Chiu, 1975). (Refer to Figure 5.2[a,b] for a display

of the bathymetry of the Loxahatchee River estuary and a river bottom profile of the

Loxahatchee River, respectively.) Estuarine conditions extend upstream from the central

embayment for roughly 2.5 km to a point where the Northwest Fork constricts to form a well-

defined river channel (McPherson and Sabanskas, 1980). This transitional area has an average

width of about 750 m with average and maximum depths of 1.25 and 3.75 m, respectively (see

Figure 5.2). This section of the Northwest Fork receives the direct outflow from the

Loxahatchee River and thus may experience large and rapid fluctuations in salinity. Upstream of

this location, salinity regimes within the Northwest Fork are more stable. Historically,

freshwater river inflows supplied by the Loxahatchee River were sufficient to maintain brackish

water conditions within this portion of the Northwest Fork, which supported diverse estuarine

fish, benthic fauna, and oyster communities in its upper reaches and marine seagrass

communities downstream near its confluence with the central embayment. Today, surface water

inflows delivered to the Northwest Fork are insufficient to restrict the upstream migration of the

salt wedge into the historical freshwater reaches of the Loxahatchee River, and hence, estuarine

conditions within this transitional area have deteriorated (Dent and Ridler, 1997).

72

Figure 5.2. (a) Bathymetry (displayed in meters below MSL) of the Loxahatchee River estuary

with river-kilometer distances plotted along the Loxahatchee River including (b) its

associated river bottom profile.

73

The North Fork is a very shallow tributary and presently contributes only a small portion

(2 percent) of the total freshwater river inflow that is delivered to the Loxahatchee River estuary

(Russell and McPherson, 1984; Sonntag and McPherson, 1984). Estuarine conditions within the

North Fork extend upstream from the central embayment for roughly 4.75 km (McPherson and

Sabanskas, 1980). The North Fork has an average width of about 250 m with average and

maximum depths of 1 and 2 m, respectively (see Figure 5.2[a]). Much of the watershed

surrounding the upstream portions of the North Fork lies within JDSP (see Figure 5.1[a]).

Nearer the central embayment, the shoreline of the North Fork is bulkheaded to support dense

residential development. Water quality is often poor due to high levels of turbidity and color

(which limits light penetration), low levels of dissolved oxygen, and occasional high

concentrations of fecal coliform bacteria (Dent et al., 1998). Management considerations

regarding the North Fork emphasize the need to improve its water-quality conditions through

improved storm water-control systems (which feed runoff to the tributary) along and stabilization

of soft organic sediments (which further add to the degradation of water quality) within the

North Fork. Further, although there is no direct control over the amount of freshwater river

inflow delivered to the North Fork, actions that can be taken to improve flushing and exchange

of water within the North Fork are encouraged as a means to provide additional improvements to

its overall water quality.

The Southwest Fork has been heavily altered, dredged, and channelized for navigational

and recreational use and to provide access to local marinas and private homes (McPherson et al.,

1982). The Southwest Fork also provides a zone for surface water inflows supplied by the C-18

canal to mix with more saline waters derived from Jupiter Inlet, which prevents these freshwater

discharges from damaging sensitive grassbeds and oyster beds located further downstream,

74

nearer the central embayment. Freshwater river inflow delivered to the Southwest Fork is

controlled by the S-46 structure to provide overflow from the C-18 canal (see Figure 1.1).

Further, operation of the S-46 structure has a significant influence on the salinity regimes

experienced within the Southwest Fork (FDEP, 1998). Estuarine conditions within the

Southwest Fork extend upstream from the central embayment for roughly 1.1 km (McPherson

and Sabanskas, 1980). The Southwest Fork has an average width of about 250 m with average

depths of 1.7 m (see Figure 5.2[a]). Freshwater discharges delivered to the Southwest Fork

provide about 21 percent of the total surface water inflows supplied to the Loxahatchee River

estuary. Periodically, due to extreme rainfall events, very large amounts of runoff from the C-18

Canal/J.W. CWMA sub-basin (see Table 5.1) are discharged into the Southwest Fork which

provides strong freshwater conditions within the Loxahatchee River estuary. In contrast, during

dry season conditions, there are long periods of time when the Southwest Fork receives no

surface water inflow from the C-18 canal.

The physical features of the Loxahatchee River estuary, namely its geomorphical

characteristics and salinity distributions, are strongly governed by the configuration of Jupiter

Inlet, coastal influences, and land-drainage alterations. A key event in the history of

hydrological changes of the Loxahatchee River estuary includes the creation of the AIW in the

early 1900s, which was constructed by dredging the connection between Lake Worth Creek and

Jupiter Sound about Jupiter Inlet (Russell and McPherson, 1984). Lake Worth Inlet was also

constructed and modifications to St. Lucie Inlet during this period further diverted surface water

inflows away from Jupiter Inlet (Vines, 1970). (Refer to Figure 5.1[a] for a map of the

Loxahatchee River watershed, which indicates the locations of these inlets with respect to Jupiter

Inlet.) As a result of such activities, the tidal prism (see footnote on page 71) increased, and an

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enhanced tidal exchange and decreased residence times of freshwater river inflows within the

Loxahatchee River estuary followed; this led to a more saline estuarine system.

Further coastal influences that have also greatly affected the hydrology of the

Loxahatchee River estuary involve those associated with the configuration of Jupiter Inlet.

Historical evidence suggests that Jupiter Inlet has been opened and closed many times in the past

due to natural conditions (DuBois, 1968). According to historical accounts, the size of oyster

shells found in prehistoric shell mounds surrounding Jupiter Inlet indicate that it must have been

open 1000 years ago. Jupiter Inlet may have been visited by Juan Ponce de Leon during his

travels down the Florida coast in 1513 (Schwartz and Ehrenberg, 2001). Pedro Menendez may

have used Jupiter Inlet in 1565 as he traveled to Cuba (Schwartz and Ehrenberg, 2001). Later,

starting in 1671, cartographers began to include Jupiter Inlet on early explorer maps (DuBois,

1968). DuBois (1968) also provides accounts given by Jonathan Dickinson, in which his journal

recalls that Jupiter Inlet was open in 1696. In 1773, a Dutch civil engineer, Bernard Romans,

related that Jupiter Inlet was closed for many years before 1769, but thereafter, he had seen it

open until 1773 (DuBois, 1968).

Many accounts taken from the nineteenth century further serve as historical evidence that

Jupiter Inlet has opened and closed periodically over its history. John Lee Williams wrote in

1837 that Jupiter Inlet had opened and closed three times within 70 years. In 1837, Jupiter Inlet

had shoaled and appeared to be closing, which it later did in 1838 shortly after the Battle of the

Loxahatchee (Courier Journal, 1988). According to the memoirs accompanying the Ive’s

Military Map of 1856, Jupiter Inlet was closed from 1840 to 1844 (DuBois, 1968). In 1844,

local citizens dug Jupiter Inlet open with shovels, after which, water flooded through and created

a channel nearly 300 m wide (Courier Journal, 1988). Jupiter Inlet stayed open until 1847, and

76

then it remained closed for the next six years. In 1853, Jupiter Inlet opened only for a short

period of time. In 1855, Major William L. Haskin of the First Artillery of the U.S. Army tried to

clear the channel, but the unusually dry season conditions provided no floodwaters to keep

Jupiter Inlet open.

Towards the turn of the century, soundings taken through Jupiter Inlet revealed depths of

2.75 and 2.25 m within the outer and inner bars, respectively; however, by the autumn of 1896,

Jupiter Inlet required reopening (DuBois, 1968). By the summer of 1901, Jupiter Inlet closed

again, but reopened a month later, as the result of actions taken by local citizens, with depths of

about 1 m within the inner and outer bars (Courier Journal, 1988). The autumn of 1910 found

Jupiter Inlet closed once again, but record high floodwaters received later in the year created a

channel of about 300 m in width (DuBois, 1968).

In response to the establishment of the JID in 1921, work to place rock for the

construction of jetties extending from Jupiter Inlet began. By 1928, the north and south jetties

extended further than 60 and 25 m, respectively, from Jupiter Inlet (Mehta et al., 1990). In 1931,

more rock was added to the jetties; however, even with the additional support given to the

channel, Jupiter Inlet continued to shoal and appeared to be closing (Cary, 1978). The channel

was dredged in 1936 and quickly closed due to shoaling within the two years following its

reopening. In 1940, two steel groins were constructed on the north side of Jupiter Inlet to stop

erosion near the shoreward side of the north jetty. In addition, a converging steel groin system

was built on the seaward side of the south jetty to increase flow velocities through Jupiter Inlet

and induce scouring between the two jetties (University of Florida, 1969). The channel was

dredged in 1941 to a depth and width of 1.8 and 20 m, respectively; however, Jupiter Inlet closed

nearly a year later (Mehta et al., 1990). From 1942 to 1947, Jupiter Inlet remained closed until

77

local citizens dredged a substantial amount of material from the closure to create a channel 2.5

and 30 m in depth and width, respectively (Cary, 1978). For many years following, the material

dredged from the channel was being deposited on the north side of Jupiter Inlet. In 1956, a 90-

meter-long concrete-capped sheet pile jetty was constructed about 30 m north of the existing

north jetty to prevent the erosion of this dredged material (Mehta et al., 1992).

Originally, Jupiter Inlet was the only outlet for the freshwaters of the Loxahatchee River,

Lake Worth Creek, and Jupiter Sound (see Figure 1.1), which provided a sufficient amount of

flow through the channel to prevent its closure. Upon the construction of Lake Worth Inlet and

modification of St. Lucie Inlet, a considerable amount of surface water inflows were diverted

away from Jupiter Inlet (Vines, 1970). (Refer to Figure 5.1[a] for a map of the Loxahatchee

River watershed, which indicates the locations of these inlets with respect to Jupiter Inlet.) As a

result, Jupiter Inlet closed more frequently and for longer periods of duration. In 1947, a regular

maintenance schedule of Jupiter Inlet was initiated by the JID, which consisted primarily of

periodic dredging activities. This schedule of regular dredging activity has since prevented

closure of the channel; however, the inherent problems of shoaling to the north and south of

Jupiter Inlet have yet to be fully resolved (Buckingham, 1984).

Overall, land-drainage alterations have rerouted surface water inflows to reduce the

effective size of the river basin and therefore total runoff (McPherson and Sabanskas, 1980).

These land-drainage alterations serve to deliver freshwater discharges to the Loxahatchee River

estuary more rapidly and abruptly, flushing the estuarine portions of the Loxahatchee River with

higher amounts of surface water inflow. During dry periods, however, drained marshes and

lowered groundwater tables are not able to provide the same historical freshwater baseflow

required to prevent upstream encroachment of saline estuarine waters (Rodis, 1973; Alexander

78

and Crook, 1974). The overall effect resulting from these land-drainage alterations has included

an estimated net loss of 10 million m3 of storage in the C-18 Canal/J.W. CWMA sub-basin (see

Table 5.1). Various proposals have been developed and actions implemented to increase the

amount of freshwater river inflow delivered to the Northwest Fork in order prevent the upstream

migration of saltwater within the Loxahatchee River (Birnhak, 1974; Federal Department of

Natural Resources, 1985); however, these increased surface water inflows proved insufficient to

substantially alter salinity conditions within the Loxahatchee River estuary.

Regions of the Loxahatchee River estuary with the highest variability of surface and

bottom salinities are presumably most responsive to changes in hydrological variables (e.g.,

those associated with tidal dynamics and river discharges). In general, surface salinity is most

dynamic within the Northwest Fork, upstream from the central embayment to a location near

Kitching Creek, while bottom salinity is most variable in the far upper reaches of the

Loxahatchee River. These spatial variations in salinity provide for stratification of the waters

within the upstream portions of the Northwest Fork; however, it is noted that there is significant

vertical mixing within the central embayment and downstream to Jupiter Inlet to overcome

stratification within these estuarine waters.

79

CHAPTER 6. PRELIMINARY MODELING EFFORTS

The following outline of the preliminary modeling efforts taken to reproduce the two-

dimensional tidal flows within the Loxahatchee River estuary involves five main sections. First,

an overview of the WNAT model domain provides information relating to previous meshing

efforts taken to discretize the spatial features of this large-scale computational domain.

Following, the development of the preliminary version of the finite element mesh is described.

Next, a section dedicated to model initialization details the boundary conditions and model

parameterizations applied throughout the present study. Preliminary model results are then

presented and discussed. Finally, simulation output obtained from a variety of model-sensitivity

runs is reviewed.

6.1. WNAT Model Domain

The WNAT model domain encompasses the Gulf of Mexico, Caribbean Sea, and northern

portion of the Atlantic Ocean found west of the 60°W meridian (Figure 6.1). The open-ocean

boundary extends from the area of Glace Bay, Nova Scotia, Canada to the vicinity of Corocora

Island in eastern Venezuela (see Figure 6.1, box 1). Bounded on the north, west, and south by

the North, Central, and South American coastlines, the WNAT model domain covers an area of

approximately 8.4 million km2. Bathymetry of the WNAT model domain ranges from zero at

the coastlines to several thousand meters in portions of the deep ocean basin.

80

Some of the major bathymetric features that influence tidal propagation through the

WNAT model domain include the continental shelf break and the edge of Blake’s Escarpment

(see Figure 6.1, box 2). Legally, the continental shelf break is declared to be located at a depth

of 183 m (Runcorn, 1967). However, southward from a point due east of North Carolina, the

slope of the sea floor along the edge of Blake’s Escarpment (near the 1200-m contour) is on the

order of 6 degrees, whereas the bathymetric gradient along the 183-m contour is on the order of 2

degrees.

Figure 6.1. Bathymetry (displayed in meters below MSL) of the WNAT model domain,

highlighting the open-ocean boundary and the areas of the continental shelf break

(183 m) and the edge of Blake’s Escarpment (1200 m) (boxes 1 and 2,

respectively).

81

Recent advances in surface water modeling have permitted the development and

successful implementation of coastal ocean circulation models for increasingly larger

computational domains (Lynch and Gray, 1979; Lynch, 1983; Kinnmark, 1985; Foreman, 1986;

Westerink and Gray, 1991; Luettich et al., 1992; Westerink et al., 1992b; Westerink et al.,

1994b; Westerink et al., 1994d; Hagen and Westerink, 1995; Luettich and Westerink, 1995;

Kolar et al., 1996). While a large-scale computational domain (e.g., the WNAT model domain;

see Figure 6.1) increases the predictive capabilities of coastal ocean models (Blain et al., 1994;

Westerink et al., 1994c), it complicates the process of computational node placement. Large-

scale computational domains require a strategic placement of computational nodes in order to

maintain acceptable levels of local and global accuracy for a given computational cost.

However, the actual meshing of larger, more complex computational domains relies on crude

criteria and results in a computational grid that is user-dependent and indirectly related to the

physics of the flow phenomena. To this end, much work has been accomplished towards

developing methods of grid generation which more successfully couple the physics (as

represented by discrete equations) underlying tidal flow and circulation to the process of

computational node placement (Hagen, 1998; Hagen et al., 2000; Hagen 2001; Hagen et al.,

2001; Hagen et al., 2002; Hagen and Parrish, 2004).

Previous meshing efforts taken to discretize the spatial features of the WNAT model

domain are presented here in chronological order for the purpose of highlighting the history

associated with the meshing of this large-scale computational domain. Following the

conclusions offered by Westerink et al. (1992b) and Westerink et al. (1994c), Roe (1998)

produced a finite element mesh for the WNAT model domain from scratch (see Table 6.1).

Mukai et al. (2002) improve the finite element mesh employed by Westerink et al. (1993) by

82

increasing its total number of computational nodes by a factor of four and through a strategic

rearrangement of these additional computational nodes (see Table 6.1). Following the findings

of Westerink et al. (1992b), Westerink et al. (1994b), and Hagen (1998), Parrish (2001) refines

the finite element mesh of Mukai et al. (2002) in the areas of the continental shelf break and the

edge of Blake’s Escarpment (see Figure 6.1, box 2), two regions of the WNAT model domain

where gradients in bathymetry are high and an increased grid resolution is required (see Table

6.1). Further details regarding the development of this finite element mesh, including its

capability to reproduce the tides within the WNAT model domain can be found in Parrish (2001)

and Parrish and Hagen (2001).

Table 6.1. Characteristics of the WNAT model domain-based finite element meshes.

Nodal spacing (km) Finite element mesha No. nodes No. elements

Minimum Maximum Boundary

Roe (1998) 32,947 61,705 8.0 32 8

Mukai et al. (2002) 254,629 492,182 1.0-4.0 25 1-4

Parrish (2001) 333,701 648,661 1.0 25 1

Kojima (2005) 47,860 89,212 0.5 160 6 a Finite element meshes are labeled according to the corresponding mesh developers/users.

Hagen et al. (2005b) perform a localized truncation error analysis (LTEA), using results

from application of the finite element mesh developed by Parrish (2001) to begin this grid

generation process; the LTEA procedure is followed with the motivation of coarsening the

overall resolution of the highly refined, finite element mesh of Parrish (2001). A series of finite

element meshes is then developed from this application of the LTEA technique, with each of the

83

following finite element meshes requiring lower levels of resolution to the describe the WNAT

model domain (Kojima, 2005). The final product of this grid generation work is shown in Figure

6.2, with the details of this LTEA-based finite element mesh listed in Table 6.1.

The effect of the LTEA technique is apparent in the resulting finite element mesh (i.e.,

the nodal spacing within the deeper waters [where little bathymetric change occurs] is relaxed

while the grid resolution over the areas of the continental shelf break and the edge of Blake’s

Escarpment [where bathymetric gradients are high] remains relatively fine; see Figure 6.2, red

box). Further, Kojima (2005) demonstrates the efficacy of this highly computationally efficient,

finite element mesh by performing an error analysis on the model results at 150 locations

scattered throughout the WNAT model domain.

Figure 6.2. LTEA-based finite element mesh of Kojima (2005), highlighting the increased grid

resolution remaining over the areas of the continental shelf break and the edge of

Blake’s Escarpment (red box).

84

6.2. Finite Element Mesh Development (Preliminary Version)

Westerink et al. (1995) have found it highly advantageous to define a computational domain

which encompasses a large expanse of the deep ocean in addition to the continental margin

region of interest (e.g., the WNAT model domain; see Figure 6.1). Therefore, a large-domain

approach is taken to ensure that the open-ocean boundary conditions are properly enforced and to

allow for the non-linear response to be generated in shallow-water regions where the tides are

known to have a more appreciable interaction with the bottom. This large-domain approach

permits for hydrodynamically simple boundary conditions to be imposed along the open-ocean

boundary, which offers three main advantages (Westerink et al., 1991; Kolar et al., 1994a;

Westerink et al., 1994c): 1) astronomical forcing is applied by coupling to global ocean models

that accurately predict the harmonic behavior of the tides in the deep ocean regions; 2) non-linear

processes in the deep ocean are insignificant; 3) a boundary located in the deep ocean is

geometrically simple. It is clear to see that the open-ocean boundary of the WNAT model

domain (as located along the 60°W meridian; see Figure 6.1, box 1) is situated in the deep ocean

waters where tidal responses vary slowly. Further, it is positioned away from any continental

shelf regions, amphidromes, or resonant areas, providing an ideal location to enforce open-ocean

boundary conditions.

Accounting for the advantages noted in the above paragraph, the Loxahatchee River

estuary is described and appended to the LTEA-based finite element mesh of Kojima (2005).

The coastline and bathymetric data used to bound and discretize the Loxahatchee River estuary

are provided by the current version of the integrated, three-dimensional estuary model (Yeh et

al., 2004; see Figure 6.3). Automatic mesh generation is accomplished through the use of the

85

Surface-water Modeling System (SMS) software package (Zundel, 2003). The resulting finite

element mesh maintains similar characteristics as the LTEA-based finite element mesh of

Kojima (2005) (see Table 6.1); however, the additional discretization required to describe the

Loxahatchee River estuary boosts the overall mesh composition to include 54,077 computational

nodes and 99,846 triangular elements (Figure 6.4[a]). The nodal density required to adequately

resolve the shoreline and bathymetric features of the Loxahatchee River estuary is in the range of

20 to 100 m (Figure 6.4[b]). (Refer to Figure 5.2[a] for a display of the bathymetry associated

with the Loxahatchee River estuary, as represented by this preliminary version of the finite

element mesh.)

Figure 6.3. Coastline and bathymetric definition of the Loxahatchee River estuary, as

represented by the current version of the integrated, three-dimensional estuary

model (after Yeh et al. [2004]).

86

Figure 6.4. Spatial discretization of the Loxahatchee River estuary: (a) finite element mesh

representation and (b) its associated nodal density (displayed in meters).

87

6.3. Model Initialization

The following model parameterizations and applied boundary conditions remain constant (except

when noted) for all tidal simulations performed herein: a spherical coordinate system is used;

tidal simulations are begun from a cold start; advective terms (see Eq. [4.4]) are not included

(except for select model-sensitivity runs); seven tidal potential forcings (K1, O1, M2, S2, N2,

K2, Q1; see Table 2.2) are applied over the interior of the computational domain; the open-ocean

boundary is depth-forced with harmonic data corresponding to these same seven tidal

constituents, as obtained from the global ocean model of Le Provost et al. (1998). In the case

that these tidal elevation data of Le Provost et al. (1998) are inaccurate (which is common in

some shallow-water regions located along the 60°W meridian; see Figure 6.1, box 1), long-term

tidal records are used to adjust the global ocean model data (Westerink et al., 1994c).

Freshwater river inflows are loaded as normal-flow boundary conditions for select model-

sensitivity runs. All mainland coastlines and island shorelines employ a zero-flux boundary

condition (similar to infinite vertical walls).

Tidal simulations are begun from the beginning of an epoch (see Appendix B); 90 days of

real time is simulated; a smooth hyperbolic tangent ramp function, which acts over 20 days, is

applied to both the tidal potential and boundary forcings (Luettich et al., 1992). A time step of 5

seconds is used to ensure that the Courant number criterion (see Eq. [4.16]) is satisfied

throughout the computational domain (Westerink et al., 1994a). Additionally, the last 45 days of

the simulated water surface elevations are harmonically analyzed (using the harmonic analysis

utility contained within ADCIRC-2DDI) in order to determine the corresponding tidal

constituents.

88

The wetting and drying algorithm is enabled (Luettich and Westerink, 1999) with the

minimum bathymetric depth set to 0.1 m (i.e., computational nodes and the accompanying

elements with water depths less than the prescribed minimum bathymetric depth are considered

to be dry). The hybrid bottom friction formulation is employed, specifying the following hybrid

bottom friction parameter values (see Eq. [4.21]) according to Hagen et al. (2005a):

; H0025.0min

=fC break = 10 m; θ = 10; γ = 1/3. (It is noted that for a variety of model-sensitivity

runs, this recommended value of is adjusted from its current setting.) Finally, horizontal

eddy viscosity (see Eq. [4.4]) is set to 5 m/s

minfC

2 and the GWCE weighting parameter (see Eq.

[4.14]) is set to 0.020 to round out the settings of the tidal simulations.

6.4. Preliminary Model Results

Two different types of comparisons are utilized in order to verify the computed water surface

elevations attained from the tidal simulations: qualitatively based, as established by visual

interpretations of tidal resynthesis plots; quantitatively based, as premised on statistical analysis

measures. Tidal resynthesis plots display 14-day resyntheses of historical and model tidal

constituents. (This 14-day time period is chosen in order to include a complete spring-neap tidal

cycle in the tidal resynthesis [see Figure 2.1].) Each tidal signal is resynthesized through the

summation of Eq. (2.4), neglecting the nodal adjustment factors in order to recreate the tides

from the beginning of an epoch (see Appendix B). All 68 (excluding the solar annual [SA] and

solar semi-annual [SSA]) tidal constituents listed in Table 2.3 are used for the resynthesis of the

historical tidal signal; the model tidal signal employs the 23 tidal constituents listed in Table 6.2.

89

Table 6.2. 23 tidal constituents employed by ADCIRC-2DDI.

Tidal constituent

Period (MSD)

Degrees persolar hour Origin

STEADY ∞ 0.0000 Principal water level

MN 27.55 0.5444 Lunar monthly constituent

SM 14.77 1.0159 Lunisolar synodic fortnightly constituent

O1 1.076 13.9430 Lunar diurnal constituent

K1 0.997 15.0411 Lunar diurnal constituent

MNS2 0.547 27.4238 Arising from interaction between MN and S2

2MS2 0.536 27.9682 Variational constituent

N2 0.527 28.4397 Larger lunar elliptic semi-diurnal constituent

M2 0.518 28.9841 Principal lunar semi-diurnal constituent

2MN2 0.508 29.5285 Smaller lunar elliptic semi-diurnal constituent

S2 0.500 30.0000 Principal solar semi-diurnal constituent

2SM2 0.484 31.0159 Shallow-water semi-diurnal constituent

MN4 0.261 57.4238 Shallow-water quarter diurnal constituent

M4 0.259 57.9682 Shallow-water overtides of principal lunar constituent

MS4 0.254 58.9841 Shallow-water quarter diurnal constituent

2MN6 0.174 86.4079 Shallow-water twelfth diurnal constituent

M6 0.173 86.9523 Shallow-water overtides of principal lunar constituent

MSN6 0.172 87.4238 Arising from interaction between M2, N2, and S2

M8 0.129 115.9364 Shallow-water eighth diurnal constituent

M10 0.104 144.9205 Shallow-water tenth diurnal constituent

P1 1.003 14.9589 Solar diurnal constituent

K2 0.499 30.0821 Lunisolar semi-diurnal constituent

Q1 1.120 13.3987 Larger lunar elliptic diurnal constituent

90

The second manner in which model results are assessed is through a statistical analysis of

the errors between the historical and model tidal signals. (It is noted that all tidal resyntheses

presented herein are resolved using a 60-second time step, providing a sufficient amount of data

to statistically analyze for the following error estimations.) Two different types of error

estimations are employed in order to more fully evaluate the sufficiency of the model to

reproduce the tides within the Loxahatchee River estuary. The first error estimation begins with

a determination of the absolute average phase error

91

Figures 6.5-6.9 display tidal resynthesis plots corresponding to the five water level

gaging stations located within the Loxahatchee River estuary (see Figure 1.1). The model tidal

signal relates to a resynthesis of the 23 tidal constituents listed in Table 6.2 as obtained from the

preliminary tidal simulations described in the preceding section on model initialization. Overall,

these preliminary model results demonstrate a good working model; however, the following

discrepancies are observed between the historical and model tidal signals presented in Figures

6.5-6.9. First, slight phasing errors are evident between the historical and model tidal signals at

Coast Guard Dock, with an apparent trend of increasing phasing error along (in the upstream

direction) the Loxahatchee River. Second, the tidal range is drastically over-predicted at all five

locations, with the model producing higher flood tide elevations and lower ebb tide elevations

with respect to the historical tidal signal.

ϕ , which is calculated by averaging the

differences between the times of cyclical peaks and troughs of the historical and model tidal

signals. (It is noted that for a semi-diurnal [M2-dominated] tide with a period of 12.4 hours, an

absolute average phase error of 10° corresponds to a time discrepancy of 20 minutes and 40

seconds.)

Figure 6.5. Resyntheses of (preliminary) model (red solid line) and historical (blue solid line) tidal constituents, corresponding to

the water level gaging station located at Coast Guard Dock.

92


the water level gaging station located at Pompano Drive.

93


the water level gaging station located at Boy Scout Dock.

94


the water level gaging station located at Kitching Creek.

95


the water level gaging station located at River Mile 9.1.

96

The model tidal signal is then adjusted for the absolute average phase error in order to

determine the goodness of fit between the historical and (phase-corrected) model tidal signals,

using the coefficient of determination as a measure of accuracy (Mendenhall and Sincich, 1994):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.1)

( )( )∑

∑−

−−= 2

22 1

HistHist

ModHistR

i

ii

where i corresponds to the time index; Histi refers to the historical tidal elevation at time i; Modi

relates to the model tidal elevation at time i; Hist is the average historical tidal elevation. A

practical interpretation of the coefficient of determination (as offered by Mendenhall and Sincich

[1994]) states that about 100(R2)% of the total sum of squares of the sample values about their

mean value (i.e., the denominator of the ratio shown in Eq. [6.1]) can be explained by (or

attributed to) using model output as a predictor. (It is noted that an R2 value of 1.00 corresponds

to a direct correlation between the historical and model tidal signals [i.e., model output describes

the historical tides without any degree of error].)

The second error estimation uses the normalized root mean square (RMS) error as a

measure of the dispersion between the historical and model tidal signals (Zwillinger, 2003):

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.2) ( )

NModHist

HistRMS ii

amp

∑ −=

21

where ampHist corresponds to the average amplitude of the historical tidal signal; N is the total

number of discrete points used in the error estimation. (It is noted that the units used to express

RMS error are the same as the units of the predicted values [i.e., in this case, model output];

97

however, normalizing the RMS error by the average amplitude of the historical tidal signal

provides a dimensionless quantity for the error estimation.) A special note regarding the

normalized RMS error is made with respect to the information that it provides about the

goodness of fit between the historical and model tidal signals. Normalized RMS error does not

require a phase correction before assessing the goodness of fit between the historical and model

tidal signals; the normalized RMS error is calculated directly (i.e., without any phase correction)

from the historical and model tidal signals through Eq. (6.2). Hence, the normalized RMS error

not only provides information relating to the goodness of fit between the historical and model

tidal signals, but also a measure of the phasing error between the two resynthesis curves.

Table 6.3 provides the errors computed using the two error estimations presented in the

above paragraphs for the tidal resynthesis plots displayed in Figures 6.5-6.9. While these errors

are presented as an example in order to introduce the two error estimations employed herein, it is

noted that the information provided in Table 6.3 is considered to be a control data set to which

further model results will be compared.

98

Table 6.3. Errors associated with the preliminary model results, in correspondence to the tidal

resynthesis plots presented in Figures 6.5-6.9.

Water level gaging stationa ϕ (°) R2 (-)b RMS (-)c

Coast Guard Dock 1.639 0.9323 0.1952

Pompano Drive 14.880 0.8263 0.3846

Boy Scout Dock 17.845 0.8368 0.3829

Kitching Creek 11.897 0.8711 0.2824

River Mile 9.1 12.560 0.8590 0.2933 a Refer to Figure 1.1 for the locations of these five water level gaging stations. b Coefficients of determination computed according to Eq. (6.1).

c Normalized RMS errors computed according to Eq. (6.2).

The features apparent in Figures 6.5-6.9 are translated in Table 6.3, with larger phasing

errors manifested in the upper portions of the Loxahatchee River estuary and a drastic over-

prediction of the tidal range at all five locations. (It is noted that the while the absolute average

phase errors and coefficients of determination reveal these phasing and goodness-of-fit features,

respectively, in a quantitative manner, the normalized RMS errors provide information relating

to both of these features through a single measure of accuracy, providing a convenient means in

which to assess further model results.) The qualitative and quantitative analyses of the

preliminary model results (see Figures 6.5-6.9 and Table 6.3, respectively) presented in the

above section suggest that a good working model has been developed; however, a need for

improvement is apparent.

99

6.5. Model-sensitivity Runs

In an attempt to improve the preliminary model results, a variety of model-sensitivity runs are

performed, modifying the simulation settings in two ways: 1) adjustments in the

parameterization of bottom friction; 2) application of (advective) freshwater river inflows. The

first set of model-sensitivity runs examines the model response due to adjustments of the

minimum bottom friction factor (see Eq. [4.21]). For this first set of model-sensitivity runs, the

model is initialized in the same manner as for the preliminary tidal simulations, with the

exception of the bottom friction parameterization, which involves changes in the minimum

bottom friction factor according to Figure 4.1 (i.e., 0055.0,0045.0,0035.0,0025.0min

=fC ).

Tables 6.4-6.6 detail the model results attained from this first set of model-sensitivity

runs. Each error estimate (e.g., absolute average phase error; coefficient of determination;

normalized RMS error) is tabulated separately in order to inter-compare the model results

obtained for the different applied values of the minimum bottom friction factor. The best

performing model results (i.e., lowest absolute average phase errors and normalized RMS errors

and highest values of the coefficient of determination) are bolded in Tables 6.4-6.6 for the

purpose of distinguishing apparent trends in the error analysis.

100

Table 6.4. Absolute average phase errors (°) associated with the first set of model-sensitivity

runs. The lowest absolute average phase errors are bolded in order to highlight the

best performing model results.


minfC Coast Guard Dock

Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 1.639 14.880 17.845 11.897 12.560

0.0035 2.861 12.550 13.734 5.806 5.882

0.0045 3.969 10.476 10.249 0.559 0.104

0.0055 4.935 8.648 7.293 3.969 4.944 a Refer to Figure 1.1 for the locations of these five water level gaging stations.

Table 6.5. Coefficients of determination (-) (see Eq. [6.1]) associated with the first set of

model-sensitivity runs. The highest values of the coefficient of determination are

bolded in order to highlight the best performing model results.



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 0.9323 0.8263 0.8368 0.8711 0.8590

0.0035 0.9378 0.8464 0.8691 0.9048 0.8901

0.0045 0.9435 0.8658 0.8951 0.9281 0.9109


101

Table 6.6. Normalized RMS errors (-) (see Eq. [6.2]) associated with the first set of model-

sensitivity runs. The lowest normalized RMS errors are bolded in order to highlight

the best performing model results.



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 0.1952 0.3846 0.3829 0.2824 0.2933

0.0035 0.1905 0.3462 0.3160 0.2200 0.2358

0.0045 0.1860 0.3117 0.2621 0.1967 0.2235


The effects of (increasing) bottom friction are noticeable through the errors presented in

Tables 6.4-6.6. Increasing bottom friction serves to resist tidal flow throughout the entire

Loxahatchee River estuary, with a more appreciable effect on the timing of the tides in the

upstream portions of the Loxahatchee River (see Table 6.4). All five locations (with the

exception of Coast Guard Dock) provide the least phasing error for the minimum bottom friction

factors, . 0055.0,0045.0min

=fC

Recall that the tidal range is drastically over-predicted for the preliminary tidal

simulations (see Figures 6.5-6.9). Increasing the minimum bottom friction factor acts to damp

the tides to levels more in line with the historical data (see Table 6.5). For all five locations, the

highest values of the coefficient of determination are attained for the minimum bottom friction

factor, . 0055.0min

=fC

102

The improvements in both the phasing and amplitude properties of the model response

are conveniently captured with the normalized RMS error (see Table 6.6). For all five locations,

the lowest normalized RMS errors are attained for the minimum bottom friction factors,

. Based on the error analysis results presented in Tables 6.4-6.6, it is

recommended that the minimum bottom friction factor,

0055.0,0045.0min

=fC

0055.0min

=fC , be held constant for the

remaining model-sensitivity runs.

The second set of model-sensitivity runs explores the sensitivity of the model to the

application of (advective) freshwater river inflows. For this second set of model-sensitivity runs,

the model is initialized in the same manner as for the preliminary tidal simulations, with the

exception of the minimum bottom friction factor, 0055.0min

=fC , and two other simulation

settings: 1) advective terms (see Eq. [4.4]) are enabled; 2) freshwater river inflows are loaded as

normal-flow boundary conditions.

Three tidal simulations are performed for this second set of model-sensitivity runs, with

the first tidal simulation employing a freshwater river inflow input that is typical of an average

wet season. An average wet season freshwater river inflow for the Loxahatchee River is

identified as 7.6 cms (SFWMD, 2002), with this quantity being divided accordingly over the four

main tributaries to the Loxahatchee River (see Figure 1.1): Lainhart Dam (3.6 cms); Cypress

Creek (3.1 cms); Hobe Grove Ditch (0.4 cms); Kitching Creek (0.5 cms). (Refer to Chapter 5 for

a presentation of the Loxahatchee River estuary, which provides information to support this

distribution of freshwater river inflows as supplied to the Loxahatchee River.) The second tidal

simulation employs a first-order of magnitude of this average wet season freshwater river inflow

103

input. The final tidal simulation serves as a control by enabling the advective terms without

applying any freshwater river inflows.

Tables 6.7-6.9 detail the model results attained from this second set of model-sensitivity

runs. Each error estimate (e.g., absolute average phase error; coefficient of determination;

normalized RMS error) is tabulated separately in order to inter-compare the model results

obtained for the different applied inputs of freshwater river inflow. The best performing model

results (i.e., lowest absolute average phase errors and normalized RMS errors and highest values

of the coefficient of determination) are bolded in Tables 6.7-6.9 for the purpose of distinguishing

apparent trends in the error analysis.

Table 6.7. Absolute average phase errors (°) associated with the second set of model-

sensitivity runs. The lowest absolute average phase errors are bolded in order to

highlight the best performing model results.

Water level gaging stationb

OrderaCoast Guard

Dock Pompano

Drive Boy Scout

Dock Kitching

Creek River Mile

9.1

Controlc 5.124 8.279 6.952 4.401 5.475

0 5.115 8.193 7.151 3.694 4.812

1 4.376 8.866 6.441 6.120 6.631 a Corresponds to the order of magnitude of the applied freshwater river inflow input. b Refer to Figure 1.1 for the locations of these five water level gaging stations.

c Corresponds to no freshwater river inflow input (i.e., enabling of the advective terms only).

104

Table 6.8. Coefficients of determination (-) (see Eq. [6.1]) associated with the second set of

model-sensitivity runs. The highest values of the coefficient of determination are



OrderaCoast Guard

Dock Pompano

Drive Boy Scout

Dock Kitching

Creek River Mile

9.1

Controlc 0.9464 0.8854 0.9175 0.9436 0.9227

0 0.9472 0.8871 0.9216 0.9288 0.8747



Table 6.9. Normalized RMS errors (-) (see Eq. [6.2]) associated with the second set of model-

sensitivity runs. The lowest normalized RMS errors are bolded in order to highlight

the best performing model results.


OrderaCoast Guard

Dock Pompano

Drive Boy Scout

Dock Kitching

Creek River Mile

9.1

Controlc 0.1878 0.2767 0.2169 0.2059 0.2463

0 0.1863 0.2754 0.2125 0.2183 0.2865



105

Inter-comparing the errors presented in Tables 6.7-6.9 (for Order = Control) to those

shown in Tables 6.4-6.6 (for ) demonstrates the insensitivity of the model to the

enabling of the advective terms. On a normalized RMS-error basis, enabling the advective terms

serves to slightly improve the model results at only two of the five water level gaging stations

(Pompano Drive, Boy Scout Dock), supporting the premise for excluding the advective terms in

all remaining tidal simulations.

0055.0min

=fC

Inter-comparing the errors presented in Tables 6.7-6.9 (for Order = 0) to those shown in

Tables 6.4-6.6 (for ) demonstrates the insensitivity of the model to the application

of (advective) freshwater river inflows. Only slight improvement, if any, is made by employing

(advective) freshwater river inflow inputs in the model runs. Further, applying a first-order of

magnitude of the average wet season freshwater river inflow input in the tidal simulations serves

to worsen the model results (on a normalized RMS-error basis) at all five locations (with the

exceptions of Coast Guard Dock and Pompano Drive) (see Table 6.9). Based on the error

analysis results presented in Tables 6.7-6.9, it is suggested that (advective) freshwater river

inflows be neglected in all remaining tidal simulations.

0055.0min

=fC

106

CHAPTER 7. DOMAIN SPECIFICATION AND

FINAL COMPUTATIONAL MESH

It is apparent from the preliminary model results presented and discussed in Chapter 6 that some

mechanism (other than bottom friction, advection, or tide/freshwater flow interaction) is

currently missing in the tidal model. With little sensitivity of the model to adjustments in the

parameterization of bottom friction and to the application of (advective) freshwater river inflows,

an additional approach is presented here, which focuses on more fully identifying the

computational domain for the present tidal model. The chapter closes with a presentation of the

final computational mesh, which is used to form the recommendations regarding the spatial

extent of the computational domain of the integrated, three-dimensional estuary model.

7.1. Finite Element Mesh Development (Second Generation)

Recall the drastically over-predicted tidal ranges reproduced in the preliminary tidal simulations

(see Figures 6.5-6.9). Increases in the minimum bottom friction factor (see Eq. [4.21]) serve to

damp the tides to levels more in line with the historical data (see Table 6.5); however, a

significant over-prediction of the tidal range still exists.

Using the knowledge gained from the work of Chiu (1975), Russell and Goodwin (1987),

and Hu (2002) on modeling the tides in the Loxahatchee River estuary (see Chapter 3, Previous

Modeling Studies for the Loxahatchee River Estuary), it may be necessary to extend the

computational domain of the present tidal model beyond its current spatial limit. From a mass-

107

balance point of view, extending the computational domain to include a larger spatial coverage

should serve to depress tidal elevations in the Loxahatchee River estuary by allowing tidal flow

to be spread over a greater area.

In particular, the AIW has been shown to have an effect on the tidal circulation occurring

within the coastal regions of the Loxahatchee River estuary (Russell and Goodwin, 1987). While

Russell and Goodwin (1987) show that the south arm of the AIW acts as a water-storage area

(providing relatively low velocities), significant velocities reproduced in the north arm of the

AIW reveal the importance of including the AIW in the computational domain. In addition,

Chiu (1975) and Hu (2002) present model results which demonstrate the enhanced tidal action

that results from increasing the hydraulic conductivity of Jupiter Inlet. Paralleling the findings of

Chiu (1975) and Hu (2002) to those of Russell and Goodwin (1987), it appears that the AIW may

also affect the hydraulic characteristics of the (coastal regions of the) Loxahatchee River estuary.

Accounting for the points noted in the above paragraphs, the computational domain of the

present tidal model is extended to include a larger spatial coverage of the AIW. Beginning with

the boundary of the preliminary version of the finite element mesh, the north and south limits of

the AIW are extended to provide a greater spatial extent of the AIW. The north arm of the AIW

is extended (roughly 78 km to the north) to include description of the AIW up to and beyond St.

Lucie and Fort Pierce Inlets; the south arm of the AIW is extended (roughly 43 km to the south)

to include description of the AIW down to and beyond Lake Worth Inlet (Figure 7.1[a]).

108

Figure 7.1. (a) Extension (black solid line) of the preliminary boundary (red solid line),

including the domain extent of the final version of the finite element mesh (dashed

inset box). The blue inset boxes relate to Figure 7.2. (b) Spatial discretization

associated with the second generation of the finite element mesh. The green inset

boxes relate to Figure 7.3.

The extended boundary is defined using USGS aerial photography as supplied by

TerraServer-USA (http://terraserver.microsoft.com/; website accessed on December 16, 2005).

The north limit of the extended boundary is defined at the entrance to the Indian River Lagoon.

Relatively narrow channels of the AIW continuing beyond the south limit of the extended

109

boundary, in addition to the entrance to the Indian River Lagoon at the north limit, provides the

basis for truncating the boundary as shown in Figure 7.1(a). (Refer to Figure 7.2 for a display of

the north and south limits of the extended boundary as overlaid on USGS aerial photography,

which depicts the entrance to the Indian River Lagoon and the relatively narrow channels

continuing to the north and south, respectively, of the extended boundary.)

Figure 7.2. The entrance to the Indian River Lagoon and the relatively narrow channels of the

AIW continuing (a) north and (b) south, respectively, of the extended boundary (red

solid line) (see blue inset boxes of Figure 7.1[a]). USGS aerial photography is

supplied by TerraServer-USA (http://terraserver.microsoft.com/; website accessed

on December 16, 2005).

110

The spatial discretization (see Figure 6.4[a]) and bathymetric definition (see Figure

5.2[a]) associated with the preliminary version of the finite element mesh remains for the second

generation of the finite element mesh. The added portions of the AIW, channels of Fort Pierce,

St. Lucie, and Lake Worth Inlets, and nearshore regions surrounding Fort Pierce, St. Lucie, and

Lake Worth Inlets are meshed using the SMS software package (Zundel, 2003) (Figure 7.1[b]).

A constant depth of 3.5 m is assigned to the added portions of the AIW. (The AIW Association

[http://www.atlintracoastal.org/index.htm; website accessed on December 19, 2005] states that

the maintenance depth of the AIW from Fort Pierce Inlet to Miami, Florida is between 3.05 and

3.65 m, justifying the use of an assumed 3.5-m depth for the added portions of the AIW.) The

channels of Fort Pierce, St. Lucie, and Lake Worth Inlets are defined according to Table 7.1

(Figure 7.3). Bathymetry in the nearshore regions surrounding Fort Pierce, St. Lucie, and Lake

Worth Inlets is derived from the LTEA-based finite element mesh of Kojima (2005) (see Figure

6.2).

Table 7.1. Hydrodynamic measurements associated with the additional inlets described by the

second generation of the finite element mesh (after Carr de Betts [1999]).

Inlet Width (m) Depth (m) Length (m) Tidal prisma (m3 710× )

Fort Pierce 270 4.2 2800 1.8

St. Lucie 470 2.6 3700 1.8

Lake Worth 290 4.0 1400 2.9 a See footnote on page 71.

111

Figure 7.3. (a,d,g) Boundary definition, (b,e,h) spatial discretization, and (c,f,i) bathymetry

(displayed in meters below MSL) associated with the second generation of the finite

element mesh, for the regions surrounding Fort Pierce, St. Lucie, and Lake Worth

Inlets, respectively (see green insets boxes of Figure 7.1[b]). USGS aerial

photography is supplied by TerraServer-USA (http://terraserver.microsoft.com/;

website accessed on December 16, 2005).

112

7.2. Improved Model Results

The second generation of the finite element mesh is then applied in a series of tidal simulations,

initializing the model in the same manner as for the preliminary tidal simulations, with the

exception of the bottom friction parameterization, which involves changes in the minimum

bottom friction factor according to Figure 4.1 (i.e., 0055.0,0045.0,0035.0,0025.0min

=fC ).

Tables 7.2-7.4 detail the model results attained from this series of model runs. Each error

estimate (e.g., absolute average phase error; coefficient of determination; normalized RMS error)

is tabulated separately in order to inter-compare the model results obtained for the different

applied values of the minimum bottom friction factor. The best performing model results (i.e.,

lowest absolute average phase errors and normalized RMS errors and highest values of the

coefficient of determination) are bolded in Tables 7.2-7.4 for the purpose of distinguishing

apparent trends in the error analysis.

No apparent trend is observed with respect to the phasing errors presented in Table 7.2;

the scatter in this phasing error is also observed through the normalized RMS errors presented in

Table 7.4. While it is difficult to determine the best performing model result on a phase- or

normalized RMS-error basis, it is evident that the minimum bottom friction factor,

, provides the best fit between the model output and historical data (see Table 7.3). 0055.0min

=fC

Tables 7.5-7.7 allow for inter-comparisons to be made between the preliminary model

results and those attained from application of the second generation of the finite element mesh

(both for ). These inter-comparisons of the model results isolate the effects

caused by extending the computational domain to include a greater extent of the AIW.

0055.0min

=fC

113

Table 7.2. Absolute average phase errors (°) associated with the application of the second

generation of the finite element mesh. The lowest absolute average phase errors are




Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 7.871 8.411 11.688 6.687 7.587

0.0035 9.462 5.768 7.559 0.796 1.175

0.0045 10.817 3.647 4.130 4.149 4.262


Table 7.3. Coefficients of determination (-) (see Eq. [6.1]) associated with the application of

the second generation of the finite element mesh. The highest values of the

coefficient of determination are bolded in order to highlight the best performing

model results.



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 0.9662 0.8987 0.9000 0.9176 0.9047

0.0035 0.9715 0.9209 0.9268 0.9412 0.9267

0.0045 0.9756 0.9378 0.9451 0.9544 0.9367


114

Table 7.4. Normalized RMS errors (-) (see Eq. [6.2]) associated with the application of the

second generation of the finite element mesh. The lowest normalized RMS errors

are bolded in order to highlight the best performing model results.



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 0.1616 0.2781 0.2887 0.2155 0.2136

0.0035 0.1636 0.2333 0.2236 0.1766 0.1993

0.0045 0.1676 0.1979 0.1787 0.1846 0.2153


Table 7.5. Absolute average phase errors (°) associated with the preliminary model runs and

application of the second generation of the finite element mesh (both for

). The lowest absolute average phase errors are bolded in order to


0055.0min

=fC


Finite element mesh Coast GuardDock

Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

Preliminary version 4.935 8.648 7.293 3.969 4.944

Second generation 11.925 1.743 1.317 8.345 8.913 a Refer to Figure 1.1 for the locations of these five water level gaging stations.

115

Table 7.6. Coefficients of determination (-) (see Eq. [6.1]) associated with the preliminary

model runs and application of the second generation of the finite element mesh

(both for ). The highest values of the coefficient of determination

are bolded in order to highlight the best performing model results.

0055.0min

=fC



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1



Table 7.7. Normalized RMS errors (-) (see Eq. [6.2]) associated with the preliminary model

runs and application of the second generation of the finite element mesh (both for

). The lowest normalized RMS errors are bolded in order to


0055.0min

=fC



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1



While Table 7.5 provides a mixed picture with respect to the timing of the tides, as

depending on the spatial coverage of the AIW, Tables 7.6 and 7.7 highlight the effects caused by

including the AIW in the computational domain. Significant improvements in the goodness of

116

fit between the model output and historical data are achieved when the AIW is included in the

computational domain (see Tables 7.6 and 7.7). These improvements in the model results (for

when the AIW is included in the computational domain) indicate that the AIW plays an

important role in the spatial distribution and timing of tidal flows occurring within the

Loxahatchee River estuary.

An interesting exploration of the residual circulation occurring through Jupiter Inlet and

the north arm of the AIW is performed in order to more fully establish the effect of the AIW on

the coastal hydrodynamics of the Loxahatchee River estuary. All residual circulation patterns

presented herein are calculated using the ATC (see Chapter 3, Tidal Asymmetry and Residual

Circulation) of a 14-day length of global velocity model output. (This 14-day time period is

chosen in order to include a complete spring-neap tidal cycle [see Figure 2.1] in the calculation

of the residual circulation.)

Recall that the ATC is defined as the average of any property as a function of tidal phase,

which is computed by dividing time-series data into sections of length equal to the period of the

M2 tidal constituent and averaging the sections (Winant and Gutierrez de Velasco, 2003).

Therefore, for all calculations (of the residual circulation) performed herein, a window (of width

equal to the period of the M2 tidal constituent) moves through a 14-day length of global velocity

model output. Within the bounds of this (M2 period-wide) moving window, the longitudinal and

latitudinal velocities, respectively, undergo the following averaging technique:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.1) ( ) ( )∑=2

2222

,,M

MMMM N

VUVU

117

The average velocities computed from Eq. (7.1) are continually stored as the (M2 period-wide)

moving window continues through the 14-day length of global velocity model output, until the

entire length of data has been analyzed. An average of the average velocities (as computed from

Eq. [7.1]) is then computed to provide the residual circulation:

( ) ( )∑−

−− =day14

22day14day14

,,N

VUVU MM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2)

Figure 7.4 displays residual circulation patterns for Jupiter Inlet and the north arm of the

AIW, as calculated from global velocity model output obtained from the preliminary model runs

and application of the second generation of the finite element mesh (both for ).

Residual circulation through Jupiter Inlet and the north arm of the AIW is dominated by a net

seaward tidal flow, with greater magnitudes of the residual circulation being concentrated in

deeper channels.

0055.0min

=fC

Much stronger outflow conditions arise for when the AIW is included in the

computational domain, further establishing the importance of the AIW on the coastal

hydrodynamics of the Loxahatchee River estuary. It is evident from Figure 7.4(a,b) that

extending the boundary of the preliminary version of the finite element mesh to describe a

greater extent of the AIW allows for more tidal flow to propagate through the AIW. Moreover,

propagation of tidal flow through the AIW is prohibited when the AIW is inadequately described

(i.e., as in the case of the preliminary version of the finite element mesh; see Figure 7.4[c,d]).

Correlating the (vast differences in the) residual circulation patterns shown in Figure 7.4 to the

(inter-comparisons of the) model results presented in Tables 7.5-7.7, it is deemed necessary to

118

include the AIW in the computational domain in order to more fully describe tidal circulation in

the Loxahatchee River estuary.

Figure 7.4. (a,c) Vectors and (b,d) magnitudes (cm/s) of the residual circulation occurring

through Jupiter Inlet and the north arm of the AIW, as based on the application of

the second generation of the finite element mesh and preliminary model runs (both

for ), respectively. 0055.0min

=fC

119

7.3. Final Computational Mesh

While including the additional coverage of the AIW in the second generation of the finite

element mesh is shown to be beneficial towards reproducing two-dimensional tidal flows within

the Loxahatchee River estuary, applying such a finely resolved finite element mesh with the

integrated, three-dimensional estuary model would be considered to be too computationally

intensive. Therefore, the final product of the modeling effort focuses on truncating the north and

south arms of the AIW at a reasonable distance from Jupiter Inlet, whereby reasonable refers to

providing enough spatial coverage of the AIW to accurately reproduce the circulation patterns

within the Loxahatchee River estuary without excessively increasing the computational

requirement for a three-dimensional estuary model.

Beginning with the second generation of the finite element mesh, the north and south

limits of the AIW are truncated at the entrances to the coastal regions surrounding St. Lucie and

Lake Worth Inlets, respectively. (Refer to Figure 7.1[a] for a dashed inset box of the domain

extent provided by the final version of the finite element mesh in relation to the boundary of the

second generation of the finite element mesh.) No meshing is required for the final version of

the finite element mesh, as all spatial discretization and bathymetric definition provided by the

second generation of the finite element mesh remains (Figure 7.5).

The final computational mesh is then applied in a series of tidal simulations, initializing

the model in the same manner as for the preliminary tidal simulations, with the exception of the

bottom friction parameterization, which involves changes in the minimum bottom friction factor

according to Figure 4.1 (i.e., ). Tables 7.8-7.10 detail the

model results attained from this series of model runs. Each error estimate (e.g., absolute average

0055.0,0045.0,0035.0,0025.0min

=fC

120

phase error; coefficient of determination; normalized RMS error) is tabulated separately in order

to inter-compare the model results obtained for the different applied values of the minimum

bottom friction factor. The best performing model results (i.e., lowest absolute average phase

errors and normalized RMS errors and highest values of the coefficient of determination) are

bolded in Tables 7.8-7.10 for the purpose of distinguishing apparent trends in the error analysis.

Figure 7.5. Final computational mesh; see Figure 7.1(a) for its domain extent in relation to the

boundary of the second generation of the finite element mesh.

121

Table 7.8. Absolute average phase errors (°) associated with the application of the final

version of the finite element mesh. The lowest absolute average phase errors are




Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 11.518 4.565 7.900 2.624 3.448

0.0035 13.346 1.591 3.467 3.514 3.088

0.0045 14.776 0.767 0.000 8.316 8.430


Table 7.9. Coefficients of determination (-) (see Eq. [6.1]) associated with the application of

the final version of the finite element mesh. The highest values of the coefficient of

determination are bolded in order to highlight the best performing model results.



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 0.9656 0.8986 0.8985 0.9162 0.9027

0.0035 0.9747 0.9297 0.9329 0.9440 0.9285

0.0045 0.9787 0.9494 0.9526 0.9560 0.9389


122

Table 7.10. Normalized RMS errors (-) (see Eq. [6.2]) associated with the application of the

final version of the finite element mesh. The lowest normalized RMS errors are




Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

0.0025 0.2067 0.2486 0.2545 0.2094 0.2256

0.0035 0.2089 0.2003 0.1939 0.1949 0.2171

0.0045 0.2128 0.1683 0.1625 0.2190 0.2470


No apparent trend is observed with respect to the phasing errors presented in Table 7.8;

the scatter in this phasing error is also observed through the normalized RMS errors presented in

Table 7.10. While it is difficult to determine the best performing model result on a phase- or

normalized RMS-error basis, it is evident that the minimum bottom friction factor,

, provides the best fit between the model output and historical data (see Table 7.9). 0055.0min

=fC

Tables 7.11-7.13 allow for inter-comparisons to be made between the improved model

results and those attained from application of the final version of the finite element mesh (both

for ). These inter-comparisons of the model results isolate the effects caused by

truncating the extended boundary of the second generation of the finite element mesh to produce

the final version of the finite element mesh.

0055.0min

=fC

123

Table 7.11. Absolute average phase errors (°) associated with the applications of the second

generation and final version of the finite element mesh (both for ).

The lowest absolute average phase errors are bolded in order to highlight the best

performing model results.

0055.0min

=fC



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1

Second generation 11.925 1.743 1.317 8.345 8.913

Final version 15.875 2.614 2.804 12.389 12.920 a Refer to Figure 1.1 for the locations of these five water level gaging stations.

Table 7.12. Coefficients of determination (-) (see Eq. [6.1]) associated with the applications of

the second generation and final version of the finite element mesh (both for

). The highest values of the coefficient of determination are bolded

in order to highlight the best performing model results.

0055.0min

=fC



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1



124

Table 7.13. Normalized RMS errors (-) (see Eq. [6.2]) associated with the applications of the

second generation and final version of the finite element mesh (both for

). The lowest normalized RMS errors are bolded in order to


0055.0min

=fC



Pompano Drive

Boy Scout Dock

Kitching Creek

River Mile 9.1



The error analysis results presented in Tables 7.11-7.13 suggest that the spatial extent (of

the AIW) provided by the final version of the finite element mesh is sufficient for the

reproduction of the tides in the Loxahatchee River estuary. In fact, on a coefficient of

determination-basis, the final version of the finite element mesh outperforms the second

generation of the finite element mesh (see Table 7.12). However, beyond an adequate

reproduction of the tidal elevations in the Loxahatchee River estuary, it is necessary to verify the

tidal circulation generated by the final version of the finite element mesh.

Figure 7.6 displays residual circulation patterns for Jupiter Inlet and the north arm of the

AIW, as calculated from global velocity model output obtained from the applications of the

second generation and final version of the finite element mesh (both for ). The

similarities in the residual circulation patterns presented in Figure 7.6 indicate that the final

version of the finite element mesh is capable of generating the tidal circulation occurring through

0055.0min

=fC

125

Jupiter Inlet and the north arm of the AIW to near the same degree as that reproduced by the

second generation of the finite element mesh.

Figure 7.6. (a,c) Vectors and (b,d) magnitudes (cm/s) of the residual circulation occurring

through Jupiter Inlet and the north arm of the AIW, as based on the applications of

the second generation and final version of the finite element mesh (both for

), respectively. 0055.0min

=fC

126

CHAPTER 8. CONCLUSIONS AND FUTURE WORK

Three variations of a finite element mesh representing the Loxahatchee River estuary and

different spatial extents of the AIW are applied in a variety of tidal simulations for the purpose of

providing: 1) recommendations for the domain extent of an integrated, surface/groundwater,

three-dimensional model; 2) nearshore, harmonically decomposed, tidal elevation boundary

conditions. A preliminary version of the finite element mesh is generated using the boundary

and bathymetric definition provided by the integrated, three-dimensional estuary model (see

Figure 6.4). Phase and amplitude errors quantified at five locations within the Loxahatchee

River estuary (see Figure 1.1) show that the limited spatial extent of the AIW offered by this

preliminary version of the finite element mesh is inadequate (see Chapter 6, Preliminary Model

Results). A calibration procedure follows with adjustments in the parameterization of bottom

friction and the application of (advective) freshwater river inflows; however, it is concluded that

some other mechanism is missing in the tidal model (see Chapter 6, Model-sensitivity Runs).

A second generation of the finite element mesh is generated by extending the boundary of

the preliminary version of the finite element mesh to include a greater spatial coverage of the

AIW, in addition to provide the description of three additional inlets (see Figure 7.1). Phase and

amplitude errors quantified at five locations within the Loxahatchee River estuary (see Figure

1.1) emphasize the importance of including the AIW in the computational domain (see Chapter 7,

Improved Model Results). Further, a significantly different pattern in the residual circulation

arises when the AIW is included in the computational domain, as opposed to that reproduced by

the application of the preliminary version of the finite element mesh (see Figure 7.4).

127

Limited by the computational requirement of the integrated, three-dimensional estuary

model, it is deemed necessary to produce a more computationally efficient version of the second

generation of the finite element mesh. This final version of the finite element mesh, which

truncates the north and south arms of the AIW at a distance from Jupiter Inlet, is shown to have

nearly the same capabilities as the second generation of the finite element mesh (see Chapter 7,

Final Computational Mesh), while providing a more reasonable run time.

Some comments are made regarding the model response when using the latter two finite

element meshes (second generation, final version). The inclusion of the additional inlets in the

computational domain (i.e., the second generation of the finite element mesh) does not appear to

have much effect on the tides in the Loxahatchee River estuary. Rather, it seems that the

additional volume (i.e., the extension of the north and south arms of the AIW) included in the

tidal model permits for better mass-conservation properties. Therefore, further improvement to

be made to the present tidal model involves the inclusion of nearby floodplains in the

computational domain. Incorporating these low-lying areas into the computational domain may

produce an effect similar to that resulting from the extension of the north and south arms of the

AIW, depressing tidal elevations in the Loxahatchee River estuary by allowing tidal flow to be

spread over a greater area.

Another enhancement to be made to the final computational mesh involves the

application of a tidal elevation forcing on the north arm of the AIW. Model output produced by

application of the second generation of the finite element mesh provides average amplitudes in

water level and velocity magnitude of 0.17 m and 0.23 m/s, respectively, at the location of the

northern (AIW) boundary of the final computational mesh. Due to these significant tidal

fluctuations and fluxes experienced through the north arm of the AIW, it is deemed appropriate

128

to impose a tidal elevation forcing on the northern (AIW) boundary of the final computational

mesh in future tidal simulations.

Normally having a limited amount of computational resources at their expense, coastal

modelers often rely on generating a computational mesh that minimizes the size of the

computational domain. This minimization of the computational domain, though, is usually

accompanied with a sacrifice in model accuracy. Further, it is common to calibrate such a model

(using a limited domain extent) to a single set of data; however, the predictive capabilities of the

model are lost when it is so strictly calibrated. The work presented in this thesis shows the

importance of exploring alternative methods of model calibration (i.e., through the identification

of the computational domain).

While much progress has been made towards the meshing of large-scale computational

domains (i.e., the WNAT model domain) (see Chapter 6, WNAT Model Domain), there exists a

scarce amount of literature related to the domain identification for localized coastal models. The

work presented in this thesis supports the need for future studies related to the identification of

the computational domain for localized coastal models. Future work regarding domain

identification in estuarine and coastal modeling could place the present study in an idealized

setting. Such a study might include the application of a variety of idealized domains in a series

of tidal simulations in order to isolate, and perhaps quantify, the effects caused by including the

additional coastal regions in the computational domain.

In closing, there is speculation that a tidally driven hydrodynamic connection exists

between all of the coastal/inlet systems found along the east coast of Florida. To this end, future

work related to the present study includes the construction (and eventual application) of a finite

element mesh which would describe the Loxahatchee River estuary and the AIW and Indian

129

River Lagoon up to and including the St. Johns River. A major modeling consideration would

include a spatially variable parameterization of bottom friction, as different vegetative

communities found along the channel bottoms of the AIW and Indian River Lagoon would

require separate characterizations of bottom roughness.

130

APPENDIX A. TIDAL POTENTIAL

131

The essential elements of a physical understanding of tidal dynamics are contained in Newton’s

Laws of Motion and the Principle of Conservation of Mass; for tidal analysis, the basic

components are Newton’s Laws of Motion and the Law of Gravitational Attraction. The

following appendix offers an elegant approach by Doodson (1921), Cartwright and Taylor

(1971), and Cartwright and Edden (1973) to formulating the tidal potential as acting on the

Earth’s surface and serves as supplementary detail to support the discussion on tidal analysis (see

Chapter 2). The mathematical development of gravitational forces and the equilibrium tide, as

presented herein, is based on potential theory and follows those derivations performed by

Doodson (1921), Cartwright and Taylor (1971), and Cartwright and Edden (1973).

The Law of Gravitational Attraction states that, for two particles of masses m1 and m2

separated by a distance r, there is a mutual force of attraction:

221

rmmGF = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.1)

where G is the universal gravitational constant. The concept of the gravitational potential of a

body is then introduced. Gravitational potential is defined as the work that must be done against

the force of gravitational attraction to remove a particle of unit mass to an infinite distance from

the body. According to potential theory, the gravitational potential at a point P on the Earth’s

surface (see Figure A.1) due to the presence of the Moon (of mass m) is given by the expression:

MPGm

P −=Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.2)

132

where this gravitational potential is a scalar property with units of L2T-2. In particular, the

gravitational force acting on a particle of unit mass is given by the gradient of the gravitational

potential, . As a simple analogy, the potential energy of a ball on a mountain depends on

its height up the mountain, but the accelerating downhill force on the ball depends on the local

slope of the ground. Applying the law of cosines to ΔOPM in Figure A.1 results in the following

manipulation:

PΩ∇−

φcos2222ararMP −+= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.3)

Therefore, the distance between point P on the Earth’s surface and the Moon is given by:

21

2

2

cos21 ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

ra

rarMP φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.4)

Hence, the distance MP may be eliminated from the gravitational potential:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.5) 21

2

2

cos21−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=Ω

ra

ra

rGm

P φ

This expression may then be expanded as a series of Legendre polynomials in increasing powers

of (a/r):

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡++++−=Ω ...coscoscos1 33

3

22

2

1 φφφ PraP

raP

ra

rGm

P . . . . . . . . . . . . . . . . . . . . . (A.6)

133

where the terms ( )φcosnP are the Legengre polynomials:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.7)

( )

( )φφ

φ

φ

cos3cos521

1cos321

cos

33

22

1

−=

−=

=

P

P

P

Figure A.1. Two-dimensional geometry of the Earth-Moon gravitational system.

The tidal forces represented by the terms in this gravitational potential are calculated

from their spatial gradients, . The first term in Eq. (A.6) is constant (except for variations

in r) and thus produces no gravitational force. The second term produces a uniform gravitational

force acting parallel to

nP∇−

OM (see Figure A.1) because differentiating Eq. (A.6) with respect to

( )φcosa yields a gradient of gravitational potential which provides the gravitational force

134

necessary to produce the acceleration in the Earth’s orbit towards the center of mass of the Earth-

Moon system. The third term of Eq. (A.6) is the major tide-producing term. For most purposes,

because (a/r) is only about (1/60), the fourth term may be neglected, as may all higher-order

terms of Eq. (A.6).

The tide-generating potential is therefore written as:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.8) ( )1cos321 2

3

2

−−=Ω φraGmP

The gravitational force acting on the unit mass at point P on the Earth’s surface may be resolved

into two components (vertically upwards and horizontally in the direction of increasing φ ,

respectively) as functions of φ :

⎟⎠⎞

⎜⎝⎛ −Δ=

∂Ω∂

−31cos2 2

1 φga

P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.9)

φφ

2sin1Δ−=∂Ω∂

− ga

P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.10)

where g is the acceleration due to gravity and Δ1 is a constant involving the masses and distances

of the celestial system (e.g., Earth-Moon, Earth-Sun). For the Earth-Moon system:

3

1 23

⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

le

l

ra

mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.11)

where ml and me are the masses of the Moon and Earth, respectively, and rl corresponds to the

Earth-Moon distance. The resulting tidal forces are shown in Figure A.2.

135

Figure A.2. (a) Vertical tidal forces, which are greatest at the equator, zero at 35° latitude, and

reversed at the poles, and (b) horizontal tidal forces, which are greatest at 45°

latitude (after Pugh [2004]).

To generalize these concepts into three-dimensions (see Figure A.3), the lunar angle φ

must be expressed in suitable astronomical variables. These are chosen to be declination of the

Moon north or south of the equator, dl; the north and south latitude of point P on the Earth’s

surface, Pφ ; and the hour angle of the Moon, C, which is the difference in longitude between the

136

meridians of point P on the Earth’s surface and sub-lunar point V on the Earth’s surface. It is

noted that the hour angle of the Moon moves through a complete cycle in 24 hours and 50

minutes, as the Earth rotates. The additional 50 minutes arises from the Moon’s own orbit

(revolving in the same direction as the Earth’s rotation with a period of 27.55 days) about Earth.

Figure A.3. Three-dimensional geometry of the Earth-Moon gravitational system.

An equilibrium tide can now be computed from the tide-generating potential (Eq. [A.8])

by replacing φcos with an expression for the changes in φ in the real situation. This expression,

which is derived from spherical trigonometry, gives:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.12) Cdd lPlP coscoscossinsincos ++= φφφ

The equilibrium tide is defined as the elevation of the sea surface that would be in equilibrium

with the tidal forces if the Earth were covered with water to such a depth that the response is

137

instantaneous (Figure A.4). While equilibrium tidal theory does not fully describe the tides as

they occur in the real oceans, the equilibrium tide serves as an important reference system for

tidal analysis (see Chapter 2). The equilibrium tide contains three coefficients that characterize

the three main tidal species: long-period, diurnal (cosC), and semi-diurnal (cos2C) tidal

frequencies.

Figure. A.4. Exaggerated equilibrium tidal ellipsoid for a water-covered Earth where the dashed

line represents the equilibrium surface under no tidal forces and the solid line

represents the equilibrium surface under tidal forces (after Knauss [1978]).

The equilibrium tide due to the presence of the Sun is expressed in a form analogous to

the lunar-induced tides, but with solar mass, the Earth-Sun distance, and the Sun’s angle of

declination substituted for the lunar parameters. It is noted that the tidal forces resulting from the

Sun are a factor of 0.46 weaker than the lunar tidal forces (see Table 2.2), because the much

greater solar mass is slightly more than offset by its greater distance from Earth.

138

APPENDIX B. NODAL CYCLES

139

The Earth’s equatorial plane is inclined at 23.45° to the plane in which the Earth orbits the Sun

(called the ecliptic). This inclination gives rise to the seasonal changes in Earth’s climate and the

regular seasonal movements of the Sun north and south of the equator. The plane in which the

Moon orbits the Earth is inclined at 5.15° to the plane of the ecliptic; this plane rotates slowly

over a period of 18.61 years. As a result, the amplitude of the lunar declination increases and

decreases slowly over this 18.61-year (nodal) period (also called an epoch). It is noted that all

tidal constituents are in phase at the beginning of an epoch, and hence, the nodal adjustment

factors (see Eq. [2.3]) are not required for a tidal resynthesis which begins at the beginning of an

epoch.

Increases in the range of lunar declination over an epoch act to decrease the amplitudes of

the semi-diurnal lunar tides. These nodal modulations decrease the average lunar semi-diurnal

equilibrium tide by 3.7 percent when the declination amplitudes are greatest, with a

corresponding 3.7 percent increase 9.305 years later. These nodal effects are apparent in long-

term records of observations, namely for locations where semi-diurnal tides dominate (see Figure

B.1).

Figure B.1. Standard deviation in the sea level variations observed at Newlyn, United Kingdom,

indicating the presence of the 18.61-year nodal modulation (after Pugh [2004]).

140

APPENDIX C. HISTORICAL WATER SURFACE ELEVATIONS AND

RESYNTHESIZED HISTORICAL TIDAL SIGNALS

141

142

Plots of historical water surface elevations and resynthesized historical tidal signals are presented

here to reveal the presence of meteorology (see footnote on page 21) in the records of

observations corresponding to the five water level gaging stations located within the interior of

the Loxahatchee River estuary (see Figure 1.1). The resynthesized historical tidal signals

corresponding to the five water level gaging stations employ all 68 (excluding the solar annual

[SA] and solar semi-annual [SSA]) tidal constituents listed in Table 2.3. Of importance, note the

local positive and negative surges contained within the overall measured signals, which are

enhanced when shown against the resynthesized historical tidal signals.

Figure C.1. Historical water surface elevations (blue solid line) plotted against the resynthesized historical tidal signal (red

solid line) for the water level gaging station located at Coast Guard Dock, corresponding to October 2003.

143


solid line) for the water level gaging station located at Coast Guard Dock, corresponding to November 2003.

144


solid line) for the water level gaging station located at Coast Guard Dock, corresponding to December 2003.

145


solid line) for the water level gaging station located at Coast Guard Dock, corresponding to January 2004.

146


solid line) for the water level gaging station located at Coast Guard Dock, corresponding to February 2004.

147


solid line) for the water level gaging station located at Coast Guard Dock, corresponding to March 2004.

148


solid line) for the water level gaging station located at Coast Guard Dock, corresponding to April 2004.

149


solid line) for the water level gaging station located at Pompano Drive, corresponding to October 2003.

150


solid line) for the water level gaging station located at Pompano Drive, corresponding to November 2003.

151


solid line) for the water level gaging station located at Pompano Drive, corresponding to December 2003.

152


solid line) for the water level gaging station located at Pompano Drive, corresponding to January 2004.

153


solid line) for the water level gaging station located at Pompano Drive, corresponding to February 2004.

154


solid line) for the water level gaging station located at Pompano Drive, corresponding to March 2004.

155


solid line) for the water level gaging station located at Pompano Drive, corresponding to April 2004.

156


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to October 2003.

157


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to November 2003.

158


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to December 2003.

159


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to January 2004.

160


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to February 2004.

161


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to March 2004.

162


solid line) for the water level gaging station located at Boy Scout Dock, corresponding to April 2004.

163


solid line) for the water level gaging station located at Kitching Creek, corresponding to October 2003.

164


solid line) for the water level gaging station located at Kitching Creek, corresponding to November 2003.

165


solid line) for the water level gaging station located at Kitching Creek, corresponding to December 2003.

166


solid line) for the water level gaging station located at Kitching Creek, corresponding to January 2004.

167


solid line) for the water level gaging station located at Kitching Creek, corresponding to February 2004.

168


solid line) for the water level gaging station located at Kitching Creek, corresponding to March 2004.

169


solid line) for the water level gaging station located at Kitching Creek, corresponding to April 2004.

170


solid line) for the water level gaging station located at River Mile 9.1, corresponding to October 2003.

171


solid line) for the water level gaging station located at River Mile 9.1, corresponding to November 2003.

172


solid line) for the water level gaging station located at River Mile 9.1, corresponding to December 2003.

173


solid line) for the water level gaging station located at River Mile 9.1, corresponding to January 2004.

174


solid line) for the water level gaging station located at River Mile 9.1, corresponding to February 2004.

175


solid line) for the water level gaging station located at River Mile 9.1, corresponding to March 2004.

176


solid line) for the water level gaging station located at River Mile 9.1, corresponding to April 2004.

177

APPENDIX D. TIDAL CONSTITUENT AMPLITUDE AND PHASE LISTING

178

179

The following listing catalogs the amplitudes and phases extracted from the harmonic analysis

presented in Chapter 2. Tidal constituents are listed in the same order as for Table 2.3 for the

five water level gaging stations located within the Loxahatchee River estuary (see Figure 1.1).

All amplitudes and phases are reported with respect to MSL and the Prime Meridian,

respectively.

Table D.1. 68 tidal constituent amplitudes and phases extracted by T-TIDE and used in the resynthesis of the historical tidal

signal.

Coast Guard Dockb Pompano Driveb Boy Scout Dockb Kitching Creekb River Mile 9.1bTidal

constituentaHn (m) gn (rad) Hn (m) gn (rad) Hn (m) gn (rad) Hn (m) gn (rad) Hn (m) gn (rad)

SA 0.0883 5.2466 0.0950 5.2506 0.1060 5.1072 0.1126 5.1646 0.1391 5.2219

SSA 0.1054 0.9084 0.1013 0.8163 0.1103 0.7718 0.1105 0.8913 0.0959 0.6896

MSM 0.0126 2.3204 0.0076 1.1093 0.0084 1.6410 0.0055 0.9660 0.0165 4.8730

MM 0.0288 1.7844 0.0271 1.6708 0.0252 1.7834 0.0309 1.6500 0.0365 1.7722

MSF 0.0092 4.5459 0.0003 3.9860 0.0045 5.7996 0.0050 5.8339 0.0060 5.6849

MF 0.0138 5.9898 0.0109 0.0093 0.0106 0.1031 0.0120 0.2215 0.0174 0.4592

ALP1 0.0007 3.2004 0.0009 5.2435 0.0008 5.1711 0.0007 5.4929 0.0007 5.6357

2Q1 0.0010 2.6810 0.0007 2.1490 0.0009 3.3250 0.0008 3.3591 0.0008 3.7073

SIG1 0.0012 4.5726 0.0012 4.8466 0.0016 5.2206 0.0019 5.3623 0.0021 5.4501

Q1 0.0096 3.9123 0.0094 4.2453 0.0091 4.3748 0.0098 4.4588 0.0115 4.4672

RHO1 0.0009 4.3539 0.0007 3.7425 0.0006 2.7868 0.0007 3.3163 0.0007 2.2544

O1 0.0488 4.0535 0.0458 4.2598 0.0456 4.3525 0.0463 4.4111 0.0471 4.4249

TAU1 0.0016 3.1383 0.0006 3.5308 0.0015 4.2871 0.0023 4.7323 0.0037 4.4616

BET1 0.0016 3.5477 0.0014 3.7942 0.0014 4.0881 0.0020 4.2651 0.0025 3.9682

NO1 0.0048 4.1843 0.0057 4.5679 0.0063 4.7209 0.0065 4.8187 0.0065 4.9400

CHI1 0.0007 5.7036 0.0005 5.4058 0.0005 1.6588 0.0003 1.9523 0.0015 1.9708

PI1 0.0028 3.8587 0.0029 4.2162 0.0031 4.6162 0.0030 4.7295 0.0041 5.0775

180



P1 0.0164 3.8608 0.0151 4.2099 0.0145 4.2612 0.0146 4.3382 0.0166 4.3248

S1 0.0089 2.3077 0.0074 2.6950 0.0041 3.1388 0.0041 3.3196 0.0071 3.2756

K1 0.0600 3.8600 0.0563 4.1019 0.0548 4.1738 0.0558 4.2237 0.0581 4.2359

PSI1 0.0041 2.8259 0.0041 2.9791 0.0035 3.1561 0.0039 3.2831 0.0029 3.9710

PHI1 0.0020 5.7357 0.0015 6.0957 0.0016 6.1973 0.0022 6.1167 0.0020 0.6552

THE1 0.0008 2.9765 0.0004 1.5615 0.0004 5.9493 0.0008 6.2251 0.0017 0.5459

J1 0.0019 3.8837 0.0014 4.6909 0.0010 5.1356 0.0012 5.4182 0.0019 5.4379

SO1 0.0010 4.8080 0.0014 5.9460 0.0018 6.0395 0.0018 6.2212 0.0023 0.0836

OO1 0.0029 4.0698 0.0026 4.4110 0.0028 4.6441 0.0025 4.6920 0.0023 4.5724

UPS1 0.0004 4.3544 0.0004 5.0318 0.0005 5.7620 0.0007 5.6880 0.0009 0.4494

OQ2 0.0015 0.1358 0.0017 5.0929 0.0006 5.5678 0.0008 5.3414 0.0025 5.9106

EPS2 0.0010 4.8719 0.0034 2.0843 0.0045 2.3813 0.0053 2.5766 0.0053 2.8175

2N2 0.0106 5.6777 0.0053 6.2135 0.0061 0.2314 0.0049 0.4644 0.0063 0.6362

MU2 0.0035 0.8629 0.0102 1.7902 0.0113 2.0413 0.0132 2.1063 0.0133 1.9211

N2 0.0689 6.1680 0.0630 0.4718 0.0638 0.6538 0.0654 0.7760 0.0691 0.8381

NU2 0.0128 6.1132 0.0136 0.1794 0.0139 0.2588 0.0127 0.3571 0.0195 0.2669

GAM2 0.0099 0.5479 0.0053 0.5604 0.0055 1.1301 0.0045 0.8449 0.0106 1.8598

H1 0.0546 1.4512 0.0499 2.0317 0.0536 2.1859 0.0545 2.1759 0.0642 2.1855

M2 0.3182 0.1518 0.2986 0.5924 0.3029 0.7524 0.3077 0.8329 0.3037 0.8508

181



H2 0.0450 5.0601 0.0510 5.4679 0.0519 5.6177 0.0478 5.7781 0.0448 5.8400

MKS2 0.0105 2.3785 0.0049 2.9712 0.0055 3.2004 0.0098 3.5320 0.0168 3.7135

LDA2 0.0047 0.1763 0.0071 0.3672 0.0086 0.4555 0.0096 0.3074 0.0147 0.3503

L2 0.0161 6.0109 0.0194 6.2310 0.0215 0.1018 0.0228 0.2046 0.0224 0.1925

T2 0.0077 5.4878 0.0088 0.0997 0.0092 0.2272 0.0084 0.3267 0.0107 0.2976

S2 0.0455 0.5986 0.0403 1.2156 0.0420 1.4586 0.0422 1.5671 0.0416 1.6394

R2 0.0059 3.0439 0.0057 3.2929 0.0062 3.6484 0.0062 3.8764 0.0047 4.0582

K2 0.0121 0.7388 0.0112 1.1174 0.0115 1.4059 0.0101 1.5261 0.0092 1.5205

MSN2 0.0003 2.5634 0.0011 3.9809 0.0021 3.9437 0.0023 4.0677 0.0033 4.4900

ETA2 0.0014 1.8900 0.0011 3.1699 0.0007 3.4446 0.0007 3.7420 0.0010 4.2110

MO3 0.0016 5.7257 0.0031 5.1461 0.0043 5.4002 0.0052 5.7278 0.0062 5.8730

M3 0.0008 0.1737 0.0013 1.4045 0.0015 2.0602 0.0015 2.6318 0.0016 2.0913

SO3 0.0003 0.7596 0.0010 3.8029 0.0008 4.1947 0.0004 4.1542 0.0005 0.5470

MK3 0.0015 5.6896 0.0027 4.6447 0.0039 4.9983 0.0048 5.4522 0.0058 5.6175

SK3 0.0009 5.4501 0.0011 5.7306 0.0015 0.0574 0.0020 0.3204 0.0024 0.5130

MN4 0.0012 4.9983 0.0068 5.9364 0.0082 6.1970 0.0070 0.0852 0.0055 0.2128

M4 0.0032 5.5987 0.0160 6.0446 0.0194 6.1954 0.0175 6.2584 0.0144 0.0051

SN4 0.0003 5.0423 0.0012 0.8802 0.0011 1.0416 0.0012 1.5980 0.0018 1.7413

MS4 0.0013 0.3119 0.0047 0.5315 0.0053 0.6969 0.0040 0.8447 0.0024 0.8952

182



MK4 0.0008 0.8903 0.0011 0.2993 0.0015 0.4857 0.0010 0.3988 0.0009 0.8894

S4 0.0005 3.7237 0.0002 5.4402 0.0001 0.8381 0.0002 4.4961 0.0003 4.3930

SK4 0.0003 6.1701 0.0004 0.8214 0.0003 1.1222 0.0003 1.6139 0.0004 2.6698

2MK5 0.0017 4.1340 0.0023 4.7414 0.0022 5.4023 0.0028 6.0303 0.0033 6.2439

2SK5 0.0004 1.9094 0.0003 3.9869 0.0003 4.0963 0.0002 4.2143 0.0003 4.9066

2MN6 0.0028 0.3639 0.0033 1.3682 0.0039 2.2944 0.0058 2.8299 0.0063 2.9168

M6 0.0053 0.4559 0.0059 1.4099 0.0069 2.2047 0.0095 2.7374 0.0097 2.8225

2MS6 0.0024 0.8814 0.0021 1.8734 0.0022 2.8669 0.0035 3.4112 0.0038 3.4479

2MK6 0.0007 1.1940 0.0005 2.0104 0.0008 3.0763 0.0012 3.4388 0.0014 3.5090

2SM6 0.0005 1.5896 0.0004 3.2592 0.0003 4.4101 0.0007 4.6628 0.0007 5.3521

MSK6 0.0003 1.3720 0.0002 2.4290 0.0002 4.0389 0.0004 4.7962 0.0004 4.6031

3MK7 0.0005 4.7014 0.0002 4.9093 0.0001 3.8350 0.0004 2.6080 0.0008 2.7700

M8 0.0012 0.5267 0.0011 0.6918 0.0022 1.4380 0.0027 1.9225 0.0022 2.0162 a Refer to Table 2.3 for a listing of the frequencies and nodal adjustment factors.

b Refer to Figure 1.1 for the locations of these five water level gaging stations.

183

APPENDIX E. COMPUTED METEOROLOGICAL RESIDUALS AND

RESYNTHESIZED SEASONAL VARIATIONS

184

185

Plots of computed meteorological residuals and resynthesized seasonal variations are presented

here to more fully uncover the presence of meteorology (see footnote on page 21) in the records

of observations corresponding to the five water level gaging stations located within the interior

of the Loxahatchee River estuary (see Figure 1.1). These plots span over a two-year time period

in order to accentuate the annual and semi-annual cyclical behavior of the meteorology (see

footnote on page 21) contained within the water level records. Meteorological residuals are

computed through Eq. (2.5) as the difference between the historical water surface elevations and

resynthesized historical tidal signals (i.e., the remaining signals may be attributed to

meteorological effects [see footnote on page 21] contained within the records of observations).

(See Appendix C for monthly plots of these historical water level data and resynthesized

historical tidal elevations.) Resynthesized seasonal variations are computed through a

resynthesis of the solar annual (SA) and solar semi-annual (SSA) tidal constituents (see Table

2.3). Correlation between the computed meteorological residuals and resynthesized seasonal

variations suggests that the observed water levels are highly influenced by long-term solar

heating and weather effects (see footnote on page 21).

Figure E.1. Computed meteorological residuals (blue solid line) plotted against the resynthesized seasonal variation (red solid

line), corresponding to the water level gaging station located at Coast Guard Dock.

186


line), corresponding to the water level gaging station located at Pompano Drive.

187


line), corresponding to the water level gaging station located at Boy Scout Dock.

188


line), corresponding to the water level gaging station located at Kitching Creek.

189

Figure E.5. Computed meteorological residuals (limited by the amount of historical water level data available; blue solid line)

plotted against the resynthesized seasonal variation (red solid line), corresponding to the water level gaging

station located at River Mile 9.1.

190

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